| L(s) = 1 | − 4·3-s + 4·5-s + 8·9-s − 16·15-s − 2·16-s + 24·17-s − 12·19-s − 16·23-s + 8·25-s − 12·27-s − 4·31-s + 4·37-s + 12·41-s − 20·43-s + 32·45-s − 16·47-s + 8·48-s − 96·51-s + 48·57-s − 4·67-s + 64·69-s − 44·71-s + 28·73-s − 32·75-s + 8·79-s − 8·80-s + 9·81-s + ⋯ |
| L(s) = 1 | − 2.30·3-s + 1.78·5-s + 8/3·9-s − 4.13·15-s − 1/2·16-s + 5.82·17-s − 2.75·19-s − 3.33·23-s + 8/5·25-s − 2.30·27-s − 0.718·31-s + 0.657·37-s + 1.87·41-s − 3.04·43-s + 4.77·45-s − 2.33·47-s + 1.15·48-s − 13.4·51-s + 6.35·57-s − 0.488·67-s + 7.70·69-s − 5.22·71-s + 3.27·73-s − 3.69·75-s + 0.900·79-s − 0.894·80-s + 81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8} \cdot 23^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8} \cdot 23^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.04986876332\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.04986876332\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + T^{4} )^{2} \) |
| 5 | \( 1 - 4 T + 8 T^{2} - 12 T^{3} + 14 T^{4} - 12 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | \( 1 + 16 T + 128 T^{2} + 816 T^{3} + 4418 T^{4} + 816 p T^{5} + 128 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| good | 3 | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + 23 T^{4} + 20 p T^{5} + 128 T^{6} + 224 T^{7} + 385 T^{8} + 224 p T^{9} + 128 p^{2} T^{10} + 20 p^{4} T^{11} + 23 p^{4} T^{12} + 4 p^{6} T^{13} + 8 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \) |
| 7 | \( 1 - 8 T^{3} + 47 T^{4} + 16 p T^{5} + 32 T^{6} + 400 T^{7} + 393 T^{8} + 400 p T^{9} + 32 p^{2} T^{10} + 16 p^{4} T^{11} + 47 p^{4} T^{12} - 8 p^{5} T^{13} + p^{8} T^{16} \) |
| 11 | \( 1 - 38 T^{2} + 559 T^{4} - 3624 T^{6} + 16613 T^{8} - 3624 p^{2} T^{10} + 559 p^{4} T^{12} - 38 p^{6} T^{14} + p^{8} T^{16} \) |
| 13 | \( 1 + 60 T^{3} - 9 p T^{4} - 480 T^{5} + 1800 T^{6} - 360 T^{7} - 14095 T^{8} - 360 p T^{9} + 1800 p^{2} T^{10} - 480 p^{3} T^{11} - 9 p^{5} T^{12} + 60 p^{5} T^{13} + p^{8} T^{16} \) |
| 17 | \( 1 - 24 T + 288 T^{2} - 2492 T^{3} + 17875 T^{4} - 109496 T^{5} + 2024 p^{2} T^{6} - 165224 p T^{7} + 12206433 T^{8} - 165224 p^{2} T^{9} + 2024 p^{4} T^{10} - 109496 p^{3} T^{11} + 17875 p^{4} T^{12} - 2492 p^{5} T^{13} + 288 p^{6} T^{14} - 24 p^{7} T^{15} + p^{8} T^{16} \) |
| 19 | \( ( 1 + 6 T + 79 T^{2} + 308 T^{3} + 2231 T^{4} + 308 p T^{5} + 79 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 29 | \( 1 - 168 T^{2} + 13172 T^{4} - 646200 T^{6} + 22104454 T^{8} - 646200 p^{2} T^{10} + 13172 p^{4} T^{12} - 168 p^{6} T^{14} + p^{8} T^{16} \) |
| 31 | \( ( 1 + 2 T + 89 T^{2} + 100 T^{3} + 3547 T^{4} + 100 p T^{5} + 89 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 37 | \( 1 - 4 T + 8 T^{2} + 204 T^{3} - 1720 T^{4} - 1420 T^{5} + 40248 T^{6} - 120124 T^{7} + 1051038 T^{8} - 120124 p T^{9} + 40248 p^{2} T^{10} - 1420 p^{3} T^{11} - 1720 p^{4} T^{12} + 204 p^{5} T^{13} + 8 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \) |
| 41 | \( ( 1 - 6 T + 159 T^{2} - 16 p T^{3} + 9557 T^{4} - 16 p^{2} T^{5} + 159 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 43 | \( 1 + 20 T + 200 T^{2} + 1020 T^{3} - 808 T^{4} - 33220 T^{5} + 17400 T^{6} + 3432980 T^{7} + 34938366 T^{8} + 3432980 p T^{9} + 17400 p^{2} T^{10} - 33220 p^{3} T^{11} - 808 p^{4} T^{12} + 1020 p^{5} T^{13} + 200 p^{6} T^{14} + 20 p^{7} T^{15} + p^{8} T^{16} \) |
| 47 | \( 1 + 16 T + 128 T^{2} + 1280 T^{3} + 16012 T^{4} + 128416 T^{5} + 824320 T^{6} + 6998448 T^{7} + 58173798 T^{8} + 6998448 p T^{9} + 824320 p^{2} T^{10} + 128416 p^{3} T^{11} + 16012 p^{4} T^{12} + 1280 p^{5} T^{13} + 128 p^{6} T^{14} + 16 p^{7} T^{15} + p^{8} T^{16} \) |
| 53 | \( 1 + 800 T^{3} + 1572 T^{4} - 40800 T^{5} + 320000 T^{6} + 712000 T^{7} - 23620442 T^{8} + 712000 p T^{9} + 320000 p^{2} T^{10} - 40800 p^{3} T^{11} + 1572 p^{4} T^{12} + 800 p^{5} T^{13} + p^{8} T^{16} \) |
| 59 | \( 1 - 364 T^{2} + 63104 T^{4} - 6721844 T^{6} + 479355310 T^{8} - 6721844 p^{2} T^{10} + 63104 p^{4} T^{12} - 364 p^{6} T^{14} + p^{8} T^{16} \) |
| 61 | \( 1 - 150 T^{2} + 15659 T^{4} - 1211524 T^{6} + 84989469 T^{8} - 1211524 p^{2} T^{10} + 15659 p^{4} T^{12} - 150 p^{6} T^{14} + p^{8} T^{16} \) |
| 67 | \( 1 + 4 T + 8 T^{2} + 1628 T^{3} + 8792 T^{4} - 15364 T^{5} + 1193400 T^{6} + 8217060 T^{7} - 18996674 T^{8} + 8217060 p T^{9} + 1193400 p^{2} T^{10} - 15364 p^{3} T^{11} + 8792 p^{4} T^{12} + 1628 p^{5} T^{13} + 8 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \) |
| 71 | \( ( 1 + 22 T + 351 T^{2} + 4092 T^{3} + 38567 T^{4} + 4092 p T^{5} + 351 p^{2} T^{6} + 22 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 73 | \( 1 - 28 T + 392 T^{2} - 3628 T^{3} + 22344 T^{4} - 100612 T^{5} + 120 p^{2} T^{6} - 95508 p T^{7} + 68649790 T^{8} - 95508 p^{2} T^{9} + 120 p^{4} T^{10} - 100612 p^{3} T^{11} + 22344 p^{4} T^{12} - 3628 p^{5} T^{13} + 392 p^{6} T^{14} - 28 p^{7} T^{15} + p^{8} T^{16} \) |
| 79 | \( ( 1 - 4 T + 106 T^{2} + 1132 T^{3} - 1082 T^{4} + 1132 p T^{5} + 106 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 83 | \( 1 + 28 T + 392 T^{2} + 3396 T^{3} + 14600 T^{4} - 4508 T^{5} - 83016 T^{6} + 6317244 T^{7} + 96756958 T^{8} + 6317244 p T^{9} - 83016 p^{2} T^{10} - 4508 p^{3} T^{11} + 14600 p^{4} T^{12} + 3396 p^{5} T^{13} + 392 p^{6} T^{14} + 28 p^{7} T^{15} + p^{8} T^{16} \) |
| 89 | \( ( 1 + 20 T + 306 T^{2} + 3340 T^{3} + 36126 T^{4} + 3340 p T^{5} + 306 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 97 | \( 1 + 8 T + 32 T^{2} + 156 T^{3} - 19525 T^{4} - 113288 T^{5} - 269336 T^{6} + 3591384 T^{7} + 219989137 T^{8} + 3591384 p T^{9} - 269336 p^{2} T^{10} - 113288 p^{3} T^{11} - 19525 p^{4} T^{12} + 156 p^{5} T^{13} + 32 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \) |
| 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
−5.53904282092374963019070348989, −5.52967217578389688643452402137, −5.51010460819668801045647775214, −5.43581181048194732211545247319, −5.05845451486696573122624407635, −4.77602732028626855617807984035, −4.58031451988114328638799025305, −4.50252590534982063385246185395, −4.33358509160324092712556202016, −4.19108150440849019984355766598, −4.10183240300884922428513727711, −3.73652424570342651401741345584, −3.63651479960187062386329050888, −3.47872696973281177661832019337, −3.17998828058368920199366373502, −2.91619472931517622440618608786, −2.82996604579959491589498542564, −2.66360423561828814651970280528, −2.29969711745704398429089627287, −1.97554563574599573788022958392, −1.69615811070278699054642596532, −1.45426937586484550235346958459, −1.38590220594716405387742798504, −1.26772954402768322173502326048, −0.083682737305110812155646850750,
0.083682737305110812155646850750, 1.26772954402768322173502326048, 1.38590220594716405387742798504, 1.45426937586484550235346958459, 1.69615811070278699054642596532, 1.97554563574599573788022958392, 2.29969711745704398429089627287, 2.66360423561828814651970280528, 2.82996604579959491589498542564, 2.91619472931517622440618608786, 3.17998828058368920199366373502, 3.47872696973281177661832019337, 3.63651479960187062386329050888, 3.73652424570342651401741345584, 4.10183240300884922428513727711, 4.19108150440849019984355766598, 4.33358509160324092712556202016, 4.50252590534982063385246185395, 4.58031451988114328638799025305, 4.77602732028626855617807984035, 5.05845451486696573122624407635, 5.43581181048194732211545247319, 5.51010460819668801045647775214, 5.52967217578389688643452402137, 5.53904282092374963019070348989
Plot not available for L-functions of degree greater than 10.
