| L(s) = 1 | + 4·2-s − 4·3-s + 8·4-s − 4·5-s − 16·6-s + 3·7-s + 18·8-s + 13·9-s − 16·10-s − 2·11-s − 32·12-s − 4·13-s + 12·14-s + 16·15-s + 34·16-s + 2·17-s + 52·18-s + 5·19-s − 32·20-s − 12·21-s − 8·22-s + 28·23-s − 72·24-s + 26·25-s − 16·26-s − 32·27-s + 24·28-s + ⋯ |
| L(s) = 1 | + 2.82·2-s − 2.30·3-s + 4·4-s − 1.78·5-s − 6.53·6-s + 1.13·7-s + 6.36·8-s + 13/3·9-s − 5.05·10-s − 0.603·11-s − 9.23·12-s − 1.10·13-s + 3.20·14-s + 4.13·15-s + 17/2·16-s + 0.485·17-s + 12.2·18-s + 1.14·19-s − 7.15·20-s − 2.61·21-s − 1.70·22-s + 5.83·23-s − 14.6·24-s + 26/5·25-s − 3.13·26-s − 6.15·27-s + 4.53·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(31^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(31^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(46.93469805\) |
| \(L(\frac12)\) |
\(\approx\) |
\(46.93469805\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 31 | \( 1 \) |
| good | 2 | \( ( 1 - p T + p T^{2} - 5 T^{3} + 11 T^{4} - 5 p T^{5} + p^{3} T^{6} - p^{4} T^{7} + p^{4} T^{8} )^{2} \) |
| 3 | \( 1 + 4 T + p T^{2} - 8 T^{3} - 4 p T^{4} - 8 T^{5} - 19 T^{6} + 22 p T^{7} + 283 T^{8} + 22 p^{2} T^{9} - 19 p^{2} T^{10} - 8 p^{3} T^{11} - 4 p^{5} T^{12} - 8 p^{5} T^{13} + p^{7} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \) |
| 5 | \( ( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} )^{4} \) |
| 7 | \( 1 - 3 T - 3 T^{2} + 34 T^{3} - 102 T^{4} + 181 T^{5} - 26 T^{6} - 264 p T^{7} + 8023 T^{8} - 264 p^{2} T^{9} - 26 p^{2} T^{10} + 181 p^{3} T^{11} - 102 p^{4} T^{12} + 34 p^{5} T^{13} - 3 p^{6} T^{14} - 3 p^{7} T^{15} + p^{8} T^{16} \) |
| 11 | \( 1 + 2 T + p T^{2} + 58 T^{3} + 116 T^{4} - 164 T^{5} + 39 p T^{6} - 4756 T^{7} - 24153 T^{8} - 4756 p T^{9} + 39 p^{3} T^{10} - 164 p^{3} T^{11} + 116 p^{4} T^{12} + 58 p^{5} T^{13} + p^{7} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) |
| 13 | \( 1 + 4 T + p T^{2} - 88 T^{3} - 532 T^{4} - 1928 T^{5} - 109 T^{6} + 23966 T^{7} + 124003 T^{8} + 23966 p T^{9} - 109 p^{2} T^{10} - 1928 p^{3} T^{11} - 532 p^{4} T^{12} - 88 p^{5} T^{13} + p^{7} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \) |
| 17 | \( 1 - 2 T - 3 T^{2} + 126 T^{3} - 532 T^{4} + 84 T^{5} + 3479 T^{6} - 28082 T^{7} + 52803 T^{8} - 28082 p T^{9} + 3479 p^{2} T^{10} + 84 p^{3} T^{11} - 532 p^{4} T^{12} + 126 p^{5} T^{13} - 3 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \) |
| 19 | \( 1 - 5 T + 29 T^{2} + 70 T^{3} - 590 T^{4} + 4435 T^{5} - 2714 T^{6} - 42900 T^{7} + 426339 T^{8} - 42900 p T^{9} - 2714 p^{2} T^{10} + 4435 p^{3} T^{11} - 590 p^{4} T^{12} + 70 p^{5} T^{13} + 29 p^{6} T^{14} - 5 p^{7} T^{15} + p^{8} T^{16} \) |
| 23 | \( ( 1 - 14 T + 113 T^{2} - 750 T^{3} + 4121 T^{4} - 750 p T^{5} + 113 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 - 10 T + 11 T^{2} + 10 p T^{3} - 2239 T^{4} + 10 p^{2} T^{5} + 11 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 + 2 T - 33 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{4} \) |
| 41 | \( 1 - 7 T + p T^{2} - 518 T^{3} + 3626 T^{4} - 15911 T^{5} + 3954 p T^{6} - 1177414 T^{7} + 5416137 T^{8} - 1177414 p T^{9} + 3954 p^{3} T^{10} - 15911 p^{3} T^{11} + 3626 p^{4} T^{12} - 518 p^{5} T^{13} + p^{7} T^{14} - 7 p^{7} T^{15} + p^{8} T^{16} \) |
| 43 | \( 1 - 6 T + 63 T^{2} + 22 T^{3} - 372 T^{4} + 19372 T^{5} - 833 p T^{6} + 91506 T^{7} + 3060523 T^{8} + 91506 p T^{9} - 833 p^{3} T^{10} + 19372 p^{3} T^{11} - 372 p^{4} T^{12} + 22 p^{5} T^{13} + 63 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) |
| 47 | \( ( 1 - 12 T + 17 T^{2} + 60 T^{3} + 961 T^{4} + 60 p T^{5} + 17 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 53 | \( 1 - 16 T + 213 T^{2} - 2088 T^{3} + 21008 T^{4} - 187488 T^{5} + 1667411 T^{6} - 13076524 T^{7} + 100516383 T^{8} - 13076524 p T^{9} + 1667411 p^{2} T^{10} - 187488 p^{3} T^{11} + 21008 p^{4} T^{12} - 2088 p^{5} T^{13} + 213 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} \) |
| 59 | \( 1 + 5 T + 69 T^{2} - 270 T^{3} - 1990 T^{4} - 37635 T^{5} - 145834 T^{6} + 485900 T^{7} + 5823099 T^{8} + 485900 p T^{9} - 145834 p^{2} T^{10} - 37635 p^{3} T^{11} - 1990 p^{4} T^{12} - 270 p^{5} T^{13} + 69 p^{6} T^{14} + 5 p^{7} T^{15} + p^{8} T^{16} \) |
| 61 | \( ( 1 - 6 T + 6 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{4} \) |
| 67 | \( ( 1 + 8 T - 3 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{4} \) |
| 71 | \( 1 + 23 T + 221 T^{2} - 18 T^{3} - 21614 T^{4} - 255201 T^{5} - 803866 T^{6} + 12228416 T^{7} + 191155087 T^{8} + 12228416 p T^{9} - 803866 p^{2} T^{10} - 255201 p^{3} T^{11} - 21614 p^{4} T^{12} - 18 p^{5} T^{13} + 221 p^{6} T^{14} + 23 p^{7} T^{15} + p^{8} T^{16} \) |
| 73 | \( 1 + 14 T + 193 T^{2} + 1162 T^{3} + 10388 T^{4} + 28532 T^{5} + 925511 T^{6} + 7337176 T^{7} + 107602183 T^{8} + 7337176 p T^{9} + 925511 p^{2} T^{10} + 28532 p^{3} T^{11} + 10388 p^{4} T^{12} + 1162 p^{5} T^{13} + 193 p^{6} T^{14} + 14 p^{7} T^{15} + p^{8} T^{16} \) |
| 79 | \( 1 - 20 T + 319 T^{2} - 2800 T^{3} + 25940 T^{4} - 208760 T^{5} + 2905241 T^{6} - 31993630 T^{7} + 344475259 T^{8} - 31993630 p T^{9} + 2905241 p^{2} T^{10} - 208760 p^{3} T^{11} + 25940 p^{4} T^{12} - 2800 p^{5} T^{13} + 319 p^{6} T^{14} - 20 p^{7} T^{15} + p^{8} T^{16} \) |
| 83 | \( 1 + 14 T + 3 T^{2} - 1578 T^{3} - 19132 T^{4} - 149148 T^{5} - 347179 T^{6} + 14598716 T^{7} + 238036263 T^{8} + 14598716 p T^{9} - 347179 p^{2} T^{10} - 149148 p^{3} T^{11} - 19132 p^{4} T^{12} - 1578 p^{5} T^{13} + 3 p^{6} T^{14} + 14 p^{7} T^{15} + p^{8} T^{16} \) |
| 89 | \( ( 1 - 20 T + 71 T^{2} + 690 T^{3} - 7699 T^{4} + 690 p T^{5} + 71 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 - 27 T + 192 T^{2} + 955 T^{3} - 23289 T^{4} + 955 p T^{5} + 192 p^{2} T^{6} - 27 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
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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
−4.52435980719603680001100598172, −4.39448573548930614357429059339, −4.29061794898971746077756115836, −3.83449799193111635304736941703, −3.81442691856251828230523749240, −3.80676860411639364076520757657, −3.74461803141583286952812632724, −3.67352087325942085493796337293, −3.36611556884902977325671845325, −3.10556657297953477120340437343, −2.91391289803254885743235969231, −2.80098343118264488974724986299, −2.73699340012843923509883138040, −2.52205750128830774953872616689, −2.49922807003681699770312221161, −2.35380088071299758194791094981, −2.26903554387706356422805868926, −1.54690774430578196714892649507, −1.50705092370514977211226580736, −1.46552546852412245497800009767, −1.07486563567061832500900629791, −1.04939137955881516980091942530, −0.944001138226368537933677893660, −0.73049204778867155993852864620, −0.56906579653739646137767397002,
0.56906579653739646137767397002, 0.73049204778867155993852864620, 0.944001138226368537933677893660, 1.04939137955881516980091942530, 1.07486563567061832500900629791, 1.46552546852412245497800009767, 1.50705092370514977211226580736, 1.54690774430578196714892649507, 2.26903554387706356422805868926, 2.35380088071299758194791094981, 2.49922807003681699770312221161, 2.52205750128830774953872616689, 2.73699340012843923509883138040, 2.80098343118264488974724986299, 2.91391289803254885743235969231, 3.10556657297953477120340437343, 3.36611556884902977325671845325, 3.67352087325942085493796337293, 3.74461803141583286952812632724, 3.80676860411639364076520757657, 3.81442691856251828230523749240, 3.83449799193111635304736941703, 4.29061794898971746077756115836, 4.39448573548930614357429059339, 4.52435980719603680001100598172
Plot not available for L-functions of degree greater than 10.
