| L(s) = 1 | − 2·3-s − 3·4-s + 8·5-s − 4·7-s + 2·9-s − 2·11-s + 6·12-s − 16·15-s + 4·16-s + 2·17-s + 10·19-s − 24·20-s + 8·21-s + 6·23-s + 38·25-s + 8·27-s + 12·28-s − 20·31-s + 4·33-s − 32·35-s − 6·36-s − 14·41-s − 2·43-s + 6·44-s + 16·45-s − 24·47-s − 8·48-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 3/2·4-s + 3.57·5-s − 1.51·7-s + 2/3·9-s − 0.603·11-s + 1.73·12-s − 4.13·15-s + 16-s + 0.485·17-s + 2.29·19-s − 5.36·20-s + 1.74·21-s + 1.25·23-s + 38/5·25-s + 1.53·27-s + 2.26·28-s − 3.59·31-s + 0.696·33-s − 5.40·35-s − 36-s − 2.18·41-s − 0.304·43-s + 0.904·44-s + 2.38·45-s − 3.50·47-s − 1.15·48-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.218081022\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.218081022\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | | \( 1 \) |
| good | 2 | $C_2^3$ | \( 1 + 3 T^{2} + 5 T^{4} + 3 p^{2} T^{6} + p^{4} T^{8} \) |
| 3 | $C_2^3$ | \( 1 + 2 T + 2 T^{2} - 8 T^{3} - 17 T^{4} - 8 p T^{5} + 2 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 7 | $C_2^2$ | \( ( 1 + 2 T - 3 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^3$ | \( 1 + 2 T + 2 T^{2} - 40 T^{3} - 161 T^{4} - 40 p T^{5} + 2 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2^3$ | \( 1 - 2 T + 2 T^{2} + 64 T^{3} - 353 T^{4} + 64 p T^{5} + 2 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2^3$ | \( 1 - 10 T + 50 T^{2} - 120 T^{3} + 239 T^{4} - 120 p T^{5} + 50 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2^3$ | \( 1 - 6 T + 18 T^{2} + 168 T^{3} - 1033 T^{4} + 168 p T^{5} + 18 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^3$ | \( 1 + 14 T + 98 T^{2} + 224 T^{3} - 113 T^{4} + 224 p T^{5} + 98 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2^3$ | \( 1 + 2 T + 2 T^{2} - 168 T^{3} - 2017 T^{4} - 168 p T^{5} + 2 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 53 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2} \) |
| 59 | $C_2^3$ | \( 1 - 14 T + 98 T^{2} + 280 T^{3} - 5441 T^{4} + 280 p T^{5} + 98 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2}( 1 - T + p T^{2} )^{2} \) |
| 67 | $C_2^3$ | \( 1 + 118 T^{2} + 9435 T^{4} + 118 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2^3$ | \( 1 - 2 T + 2 T^{2} + 280 T^{3} - 5321 T^{4} + 280 p T^{5} + 2 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2^2$ | \( ( 1 - 46 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 154 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 89 | $C_2^3$ | \( 1 + 10 T + 50 T^{2} - 1280 T^{3} - 14321 T^{4} - 1280 p T^{5} + 50 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2^3$ | \( 1 + 190 T^{2} + 26691 T^{4} + 190 p^{2} T^{6} + p^{4} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.29592278342521876728442913755, −6.78316521649286718765728522410, −6.67554035218004868578148928744, −6.65072733715552116915886392067, −6.63636174915279763552136641533, −5.83915243806368295595526061331, −5.79590298353089677351657829263, −5.69882536936759048747690989938, −5.44371472571557917579264891623, −5.22899791923221364685780287336, −5.06060529110031734913527218789, −5.00717456959150086359738553808, −4.97547141097405801596249248058, −4.17824407525146083743111337406, −3.96010405808115085474465899347, −3.45951397226406188242245968296, −3.42986486708478238690131639231, −3.18805179587204126384863910430, −2.58920020913594767061340126830, −2.41332692274526104745685199900, −2.35962117803260532487423744584, −1.41289348664036284252441248091, −1.33048370382117695537351313446, −1.16369892729371021552019497675, −0.32011960718549032532015633729,
0.32011960718549032532015633729, 1.16369892729371021552019497675, 1.33048370382117695537351313446, 1.41289348664036284252441248091, 2.35962117803260532487423744584, 2.41332692274526104745685199900, 2.58920020913594767061340126830, 3.18805179587204126384863910430, 3.42986486708478238690131639231, 3.45951397226406188242245968296, 3.96010405808115085474465899347, 4.17824407525146083743111337406, 4.97547141097405801596249248058, 5.00717456959150086359738553808, 5.06060529110031734913527218789, 5.22899791923221364685780287336, 5.44371472571557917579264891623, 5.69882536936759048747690989938, 5.79590298353089677351657829263, 5.83915243806368295595526061331, 6.63636174915279763552136641533, 6.65072733715552116915886392067, 6.67554035218004868578148928744, 6.78316521649286718765728522410, 7.29592278342521876728442913755
