| L(s) = 1 | − 2-s + 4·3-s + 4-s − 4·6-s + 7-s + 6·9-s − 8·11-s + 4·12-s − 14·13-s − 14-s − 16-s + 6·17-s − 6·18-s − 3·19-s + 4·21-s + 8·22-s − 2·23-s + 14·26-s + 28-s + 32·29-s − 9·31-s − 32-s − 32·33-s − 6·34-s + 6·36-s − 8·37-s + 3·38-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 2.30·3-s + 1/2·4-s − 1.63·6-s + 0.377·7-s + 2·9-s − 2.41·11-s + 1.15·12-s − 3.88·13-s − 0.267·14-s − 1/4·16-s + 1.45·17-s − 1.41·18-s − 0.688·19-s + 0.872·21-s + 1.70·22-s − 0.417·23-s + 2.74·26-s + 0.188·28-s + 5.94·29-s − 1.61·31-s − 0.176·32-s − 5.57·33-s − 1.02·34-s + 36-s − 1.31·37-s + 0.486·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3382663750\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3382663750\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( ( 1 - T + T^{2} )^{4} \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T + 11 T^{2} - 12 T^{3} + 116 T^{4} - 12 p T^{5} + 11 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| good | 2 | \( 1 + T - T^{3} + 3 T^{5} + 7 T^{6} + p^{3} T^{7} - 3 p T^{8} + p^{4} T^{9} + 7 p^{2} T^{10} + 3 p^{3} T^{11} - p^{5} T^{13} + p^{7} T^{15} + p^{8} T^{16} \) |
| 11 | \( 1 + 8 T + 28 T^{2} + 120 T^{3} + 402 T^{4} + 36 p T^{5} + 384 T^{6} - 2384 T^{7} - 29421 T^{8} - 2384 p T^{9} + 384 p^{2} T^{10} + 36 p^{4} T^{11} + 402 p^{4} T^{12} + 120 p^{5} T^{13} + 28 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \) |
| 13 | \( ( 1 + 7 T + 33 T^{2} + 94 T^{3} + 326 T^{4} + 94 p T^{5} + 33 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 17 | \( 1 - 6 T - 12 T^{2} + 108 T^{3} + 22 T^{4} + 258 T^{5} - 5760 T^{6} - 6186 T^{7} + 134727 T^{8} - 6186 p T^{9} - 5760 p^{2} T^{10} + 258 p^{3} T^{11} + 22 p^{4} T^{12} + 108 p^{5} T^{13} - 12 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) |
| 19 | \( 1 + 3 T - 30 T^{2} + 125 T^{3} + 51 p T^{4} - 3616 T^{5} + 6406 T^{6} + 66486 T^{7} - 238148 T^{8} + 66486 p T^{9} + 6406 p^{2} T^{10} - 3616 p^{3} T^{11} + 51 p^{5} T^{12} + 125 p^{5} T^{13} - 30 p^{6} T^{14} + 3 p^{7} T^{15} + p^{8} T^{16} \) |
| 23 | \( 1 + 2 T - 20 T^{2} - 84 T^{3} - 562 T^{4} - 922 T^{5} + 2568 T^{6} + 43258 T^{7} + 424639 T^{8} + 43258 p T^{9} + 2568 p^{2} T^{10} - 922 p^{3} T^{11} - 562 p^{4} T^{12} - 84 p^{5} T^{13} - 20 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) |
| 29 | \( ( 1 - 4 T + p T^{2} )^{8} \) |
| 31 | \( 1 + 9 T + 13 T^{2} + 90 T^{3} + 749 T^{4} - 1629 T^{5} + 22230 T^{6} + 154161 T^{7} - 324118 T^{8} + 154161 p T^{9} + 22230 p^{2} T^{10} - 1629 p^{3} T^{11} + 749 p^{4} T^{12} + 90 p^{5} T^{13} + 13 p^{6} T^{14} + 9 p^{7} T^{15} + p^{8} T^{16} \) |
| 37 | \( 1 + 8 T - 22 T^{2} - 800 T^{3} - 2955 T^{4} + 21128 T^{5} + 190466 T^{6} - 114864 T^{7} - 6645332 T^{8} - 114864 p T^{9} + 190466 p^{2} T^{10} + 21128 p^{3} T^{11} - 2955 p^{4} T^{12} - 800 p^{5} T^{13} - 22 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \) |
| 41 | \( ( 1 - 4 T + 104 T^{2} - 256 T^{3} + 4966 T^{4} - 256 p T^{5} + 104 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 - 5 T + 120 T^{2} - 677 T^{3} + 6782 T^{4} - 677 p T^{5} + 120 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 47 | \( 1 + 6 T - 88 T^{2} + 52 T^{3} + 6918 T^{4} - 17418 T^{5} - 221392 T^{6} + 441630 T^{7} + 3733263 T^{8} + 441630 p T^{9} - 221392 p^{2} T^{10} - 17418 p^{3} T^{11} + 6918 p^{4} T^{12} + 52 p^{5} T^{13} - 88 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} \) |
| 53 | \( 1 - 6 T - 100 T^{2} + 508 T^{3} + 4614 T^{4} - 6198 T^{5} - 355984 T^{6} - 202434 T^{7} + 25854567 T^{8} - 202434 p T^{9} - 355984 p^{2} T^{10} - 6198 p^{3} T^{11} + 4614 p^{4} T^{12} + 508 p^{5} T^{13} - 100 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) |
| 59 | \( 1 + 10 T - 96 T^{2} - 1492 T^{3} + 4998 T^{4} + 116970 T^{5} + 158320 T^{6} - 3153670 T^{7} - 19452177 T^{8} - 3153670 p T^{9} + 158320 p^{2} T^{10} + 116970 p^{3} T^{11} + 4998 p^{4} T^{12} - 1492 p^{5} T^{13} - 96 p^{6} T^{14} + 10 p^{7} T^{15} + p^{8} T^{16} \) |
| 61 | \( ( 1 - 5 T - 36 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{4} \) |
| 67 | \( 1 + T - 250 T^{2} - 133 T^{3} + 38093 T^{4} + 11624 T^{5} - 3926238 T^{6} - 284754 T^{7} + 306049684 T^{8} - 284754 p T^{9} - 3926238 p^{2} T^{10} + 11624 p^{3} T^{11} + 38093 p^{4} T^{12} - 133 p^{5} T^{13} - 250 p^{6} T^{14} + p^{7} T^{15} + p^{8} T^{16} \) |
| 71 | \( ( 1 - 22 T + 252 T^{2} - 1806 T^{3} + 12966 T^{4} - 1806 p T^{5} + 252 p^{2} T^{6} - 22 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 73 | \( 1 + 4 T - 154 T^{2} + 776 T^{3} + 16497 T^{4} - 112424 T^{5} - 312970 T^{6} + 5766588 T^{7} - 11491964 T^{8} + 5766588 p T^{9} - 312970 p^{2} T^{10} - 112424 p^{3} T^{11} + 16497 p^{4} T^{12} + 776 p^{5} T^{13} - 154 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \) |
| 79 | \( 1 + 8 T - 182 T^{2} - 1560 T^{3} + 19865 T^{4} + 148676 T^{5} - 1461174 T^{6} - 5521756 T^{7} + 104973092 T^{8} - 5521756 p T^{9} - 1461174 p^{2} T^{10} + 148676 p^{3} T^{11} + 19865 p^{4} T^{12} - 1560 p^{5} T^{13} - 182 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \) |
| 83 | \( ( 1 - 2 T + 220 T^{2} - 894 T^{3} + 22430 T^{4} - 894 p T^{5} + 220 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 89 | \( 1 - 10 T - 260 T^{2} + 1572 T^{3} + 55478 T^{4} - 195874 T^{5} - 7285344 T^{6} + 5340058 T^{7} + 785732359 T^{8} + 5340058 p T^{9} - 7285344 p^{2} T^{10} - 195874 p^{3} T^{11} + 55478 p^{4} T^{12} + 1572 p^{5} T^{13} - 260 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} \) |
| 97 | \( ( 1 + 12 T + 330 T^{2} + 3296 T^{3} + 45123 T^{4} + 3296 p T^{5} + 330 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.82840869597594237943751875395, −4.59430404471440468687552650590, −4.52193609994765675411079325334, −4.51894101028242584156238998031, −4.22316598259996685099275378338, −3.94691966640682643247003402357, −3.70941787362599407970629209820, −3.69062112691467246102767606929, −3.68567419831959747848901354704, −3.33318813516674120268379882559, −3.20276399680630558014620456670, −2.93317867844550433981922580493, −2.84027558550633241430122622834, −2.69515125301065187998131578116, −2.64860545439507183264863186063, −2.42224407478160468702517585864, −2.40242824769786116720638902375, −2.34551041404329827755238641505, −2.04055388386668836136929057834, −1.94889632303929681345160572334, −1.58127863030335264267282816148, −1.28345399016261397751200899829, −0.941099163757655074778257927121, −0.73750735813876108032058079632, −0.091229220869531626863596891657,
0.091229220869531626863596891657, 0.73750735813876108032058079632, 0.941099163757655074778257927121, 1.28345399016261397751200899829, 1.58127863030335264267282816148, 1.94889632303929681345160572334, 2.04055388386668836136929057834, 2.34551041404329827755238641505, 2.40242824769786116720638902375, 2.42224407478160468702517585864, 2.64860545439507183264863186063, 2.69515125301065187998131578116, 2.84027558550633241430122622834, 2.93317867844550433981922580493, 3.20276399680630558014620456670, 3.33318813516674120268379882559, 3.68567419831959747848901354704, 3.69062112691467246102767606929, 3.70941787362599407970629209820, 3.94691966640682643247003402357, 4.22316598259996685099275378338, 4.51894101028242584156238998031, 4.52193609994765675411079325334, 4.59430404471440468687552650590, 4.82840869597594237943751875395
Plot not available for L-functions of degree greater than 10.
