| L(s) = 1 | + 2·5-s + 7-s − 5·11-s − 3·13-s − 2·17-s − 8·19-s − 2·23-s + 8·25-s + 2·29-s + 2·35-s + 4·37-s + 3·41-s − 5·43-s + 30·47-s − 9·49-s + 24·53-s − 10·55-s + 20·59-s + 24·61-s − 6·65-s + 14·67-s + 22·71-s − 10·73-s − 5·77-s − 35·83-s − 4·85-s + 18·89-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 0.377·7-s − 1.50·11-s − 0.832·13-s − 0.485·17-s − 1.83·19-s − 0.417·23-s + 8/5·25-s + 0.371·29-s + 0.338·35-s + 0.657·37-s + 0.468·41-s − 0.762·43-s + 4.37·47-s − 9/7·49-s + 3.29·53-s − 1.34·55-s + 2.60·59-s + 3.07·61-s − 0.744·65-s + 1.71·67-s + 2.61·71-s − 1.17·73-s − 0.569·77-s − 3.84·83-s − 0.433·85-s + 1.90·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{32} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{32} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(9.220379538\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.220379538\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T + 10 T^{2} - 5 T^{3} + 101 T^{4} - 5 p T^{5} + 10 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| good | 5 | \( 1 - 2 T - 4 T^{2} + 26 T^{3} - 7 p T^{4} - 9 p T^{5} + 7 p^{2} T^{6} - 167 T^{7} - 181 T^{8} - 167 p T^{9} + 7 p^{4} T^{10} - 9 p^{4} T^{11} - 7 p^{5} T^{12} + 26 p^{5} T^{13} - 4 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \) |
| 11 | \( 1 + 5 T - 13 T^{2} - 68 T^{3} + 232 T^{4} + 651 T^{5} - 2804 T^{6} - 5056 T^{7} + 15623 T^{8} - 5056 p T^{9} - 2804 p^{2} T^{10} + 651 p^{3} T^{11} + 232 p^{4} T^{12} - 68 p^{5} T^{13} - 13 p^{6} T^{14} + 5 p^{7} T^{15} + p^{8} T^{16} \) |
| 13 | \( 1 + 3 T - 31 T^{2} - 82 T^{3} + 46 p T^{4} + 1159 T^{5} - 8178 T^{6} - 7570 T^{7} + 96573 T^{8} - 7570 p T^{9} - 8178 p^{2} T^{10} + 1159 p^{3} T^{11} + 46 p^{5} T^{12} - 82 p^{5} T^{13} - 31 p^{6} T^{14} + 3 p^{7} T^{15} + p^{8} T^{16} \) |
| 17 | \( 1 + 2 T - 31 T^{2} - 98 T^{3} + 256 T^{4} + 1164 T^{5} - 3605 T^{6} - 2620 T^{7} + 123407 T^{8} - 2620 p T^{9} - 3605 p^{2} T^{10} + 1164 p^{3} T^{11} + 256 p^{4} T^{12} - 98 p^{5} T^{13} - 31 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) |
| 19 | \( 1 + 8 T + 18 T^{2} - 122 T^{3} - 1183 T^{4} - 5499 T^{5} - 5597 T^{6} + 87515 T^{7} + 636201 T^{8} + 87515 p T^{9} - 5597 p^{2} T^{10} - 5499 p^{3} T^{11} - 1183 p^{4} T^{12} - 122 p^{5} T^{13} + 18 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \) |
| 23 | \( 1 + 2 T - 34 T^{2} + 214 T^{3} + 47 p T^{4} - 6213 T^{5} + 29101 T^{6} + 6547 p T^{7} - 804883 T^{8} + 6547 p^{2} T^{9} + 29101 p^{2} T^{10} - 6213 p^{3} T^{11} + 47 p^{5} T^{12} + 214 p^{5} T^{13} - 34 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) |
| 29 | \( 1 - 2 T - 52 T^{2} + 200 T^{3} + 1231 T^{4} - 6888 T^{5} + 20272 T^{6} + 115210 T^{7} - 1317640 T^{8} + 115210 p T^{9} + 20272 p^{2} T^{10} - 6888 p^{3} T^{11} + 1231 p^{4} T^{12} + 200 p^{5} T^{13} - 52 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \) |
| 31 | \( ( 1 + 49 T^{2} + 340 T^{3} + 693 T^{4} + 340 p T^{5} + 49 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( 1 - 4 T - 102 T^{2} + 226 T^{3} + 6515 T^{4} - 5991 T^{5} - 316301 T^{6} + 100151 T^{7} + 12334311 T^{8} + 100151 p T^{9} - 316301 p^{2} T^{10} - 5991 p^{3} T^{11} + 6515 p^{4} T^{12} + 226 p^{5} T^{13} - 102 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \) |
| 41 | \( 1 - 3 T - 83 T^{2} - 78 T^{3} + 4228 T^{4} + 10467 T^{5} - 55460 T^{6} - 313824 T^{7} - 699287 T^{8} - 313824 p T^{9} - 55460 p^{2} T^{10} + 10467 p^{3} T^{11} + 4228 p^{4} T^{12} - 78 p^{5} T^{13} - 83 p^{6} T^{14} - 3 p^{7} T^{15} + p^{8} T^{16} \) |
| 43 | \( 1 + 5 T - 117 T^{2} - 392 T^{3} + 8696 T^{4} + 14805 T^{5} - 526994 T^{6} - 217624 T^{7} + 26134839 T^{8} - 217624 p T^{9} - 526994 p^{2} T^{10} + 14805 p^{3} T^{11} + 8696 p^{4} T^{12} - 392 p^{5} T^{13} - 117 p^{6} T^{14} + 5 p^{7} T^{15} + p^{8} T^{16} \) |
| 47 | \( ( 1 - 15 T + 233 T^{2} - 2034 T^{3} + 17403 T^{4} - 2034 p T^{5} + 233 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 53 | \( 1 - 24 T + 292 T^{2} - 2262 T^{3} + 9703 T^{4} + 15291 T^{5} - 746723 T^{6} + 8701131 T^{7} - 71884493 T^{8} + 8701131 p T^{9} - 746723 p^{2} T^{10} + 15291 p^{3} T^{11} + 9703 p^{4} T^{12} - 2262 p^{5} T^{13} + 292 p^{6} T^{14} - 24 p^{7} T^{15} + p^{8} T^{16} \) |
| 59 | \( ( 1 - 10 T + 146 T^{2} - 1365 T^{3} + 12669 T^{4} - 1365 p T^{5} + 146 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - 12 T + 139 T^{2} - 602 T^{3} + 5019 T^{4} - 602 p T^{5} + 139 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 - 7 T + 16 T^{2} - 437 T^{3} + 5693 T^{4} - 437 p T^{5} + 16 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 - 11 T + 143 T^{2} - 48 T^{3} + 2421 T^{4} - 48 p T^{5} + 143 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 73 | \( 1 + 10 T - 102 T^{2} - 870 T^{3} + 5967 T^{4} - 10365 T^{5} - 932083 T^{6} + 1498025 T^{7} + 95164587 T^{8} + 1498025 p T^{9} - 932083 p^{2} T^{10} - 10365 p^{3} T^{11} + 5967 p^{4} T^{12} - 870 p^{5} T^{13} - 102 p^{6} T^{14} + 10 p^{7} T^{15} + p^{8} T^{16} \) |
| 79 | \( ( 1 + 196 T^{2} - 227 T^{3} + 19215 T^{4} - 227 p T^{5} + 196 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 83 | \( 1 + 35 T + 509 T^{2} + 4930 T^{3} + 55276 T^{4} + 684585 T^{5} + 6675268 T^{6} + 54247070 T^{7} + 460542995 T^{8} + 54247070 p T^{9} + 6675268 p^{2} T^{10} + 684585 p^{3} T^{11} + 55276 p^{4} T^{12} + 4930 p^{5} T^{13} + 509 p^{6} T^{14} + 35 p^{7} T^{15} + p^{8} T^{16} \) |
| 89 | \( 1 - 18 T + 94 T^{2} - 2358 T^{3} + 30313 T^{4} - 48465 T^{5} + 1503889 T^{6} - 14840001 T^{7} - 52709237 T^{8} - 14840001 p T^{9} + 1503889 p^{2} T^{10} - 48465 p^{3} T^{11} + 30313 p^{4} T^{12} - 2358 p^{5} T^{13} + 94 p^{6} T^{14} - 18 p^{7} T^{15} + p^{8} T^{16} \) |
| 97 | \( 1 + 19 T + 27 T^{2} - 1152 T^{3} - 1014 T^{4} + 44937 T^{5} - 376420 T^{6} - 10889992 T^{7} - 133631955 T^{8} - 10889992 p T^{9} - 376420 p^{2} T^{10} + 44937 p^{3} T^{11} - 1014 p^{4} T^{12} - 1152 p^{5} T^{13} + 27 p^{6} T^{14} + 19 p^{7} T^{15} + p^{8} T^{16} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.82696633548746711904032426791, −3.74889429035135168679811446843, −3.57461935842176899438377717784, −3.39982896216240104976363968788, −3.25408633007624470159924726668, −3.04730537932945037332534801937, −2.97922602932150823943701807127, −2.82649544800781749307218446829, −2.62486321744597541476811208730, −2.58511098859180803642265077325, −2.57613757866788938472460471510, −2.26732185588627795411853249342, −2.26025873445142527310847620104, −2.20129936069426071403403384480, −2.09609242063711890105776046672, −2.00962439423830129909738996389, −1.87214234157232524726099838967, −1.47010181586565327534541415974, −1.24863137524508205091808167725, −1.08725429476853842883143492877, −1.02796936047332067314023437245, −0.987803480648340286507113111412, −0.46345655423000142410485629197, −0.41921574658519278408244036335, −0.33146591072060909785923568187,
0.33146591072060909785923568187, 0.41921574658519278408244036335, 0.46345655423000142410485629197, 0.987803480648340286507113111412, 1.02796936047332067314023437245, 1.08725429476853842883143492877, 1.24863137524508205091808167725, 1.47010181586565327534541415974, 1.87214234157232524726099838967, 2.00962439423830129909738996389, 2.09609242063711890105776046672, 2.20129936069426071403403384480, 2.26025873445142527310847620104, 2.26732185588627795411853249342, 2.57613757866788938472460471510, 2.58511098859180803642265077325, 2.62486321744597541476811208730, 2.82649544800781749307218446829, 2.97922602932150823943701807127, 3.04730537932945037332534801937, 3.25408633007624470159924726668, 3.39982896216240104976363968788, 3.57461935842176899438377717784, 3.74889429035135168679811446843, 3.82696633548746711904032426791
Plot not available for L-functions of degree greater than 10.
