| L(s) = 1 | + 2·3-s + 4·4-s + 2·7-s − 3·9-s − 6·11-s + 8·12-s + 12·16-s + 4·21-s − 25-s − 14·27-s + 8·28-s − 12·33-s − 12·36-s + 2·37-s − 6·41-s − 24·44-s − 6·47-s + 24·48-s − 11·49-s − 18·53-s − 6·63-s + 32·64-s + 8·67-s + 18·71-s + 14·73-s − 2·75-s − 12·77-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 2·4-s + 0.755·7-s − 9-s − 1.80·11-s + 2.30·12-s + 3·16-s + 0.872·21-s − 1/5·25-s − 2.69·27-s + 1.51·28-s − 2.08·33-s − 2·36-s + 0.328·37-s − 0.937·41-s − 3.61·44-s − 0.875·47-s + 3.46·48-s − 1.57·49-s − 2.47·53-s − 0.755·63-s + 4·64-s + 0.977·67-s + 2.13·71-s + 1.63·73-s − 0.230·75-s − 1.36·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 34225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 34225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.511703995\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.511703995\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 37 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 158 T^{2} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.63004223667296844304306305156, −12.50358088755231677444729855335, −11.54785739355816888989907261646, −11.44896624550515594720021122212, −10.83169012951517833626238175733, −10.76889219441319236085810401514, −9.785026238016951011732052367999, −9.542465825427745594614941281933, −8.322574005098618135314836185926, −8.313925175787532460037677305061, −7.70870743398779290984098628138, −7.63106118534999676478252237602, −6.46044894624900883795784604601, −6.28353673742342639585453444603, −5.23124869122615795084860724530, −5.13397029034002838170016067879, −3.33397361971881193345304846920, −3.25662461644819035987483667076, −2.28967220750736277883834326003, −2.02920723204218293176594689595,
2.02920723204218293176594689595, 2.28967220750736277883834326003, 3.25662461644819035987483667076, 3.33397361971881193345304846920, 5.13397029034002838170016067879, 5.23124869122615795084860724530, 6.28353673742342639585453444603, 6.46044894624900883795784604601, 7.63106118534999676478252237602, 7.70870743398779290984098628138, 8.313925175787532460037677305061, 8.322574005098618135314836185926, 9.542465825427745594614941281933, 9.785026238016951011732052367999, 10.76889219441319236085810401514, 10.83169012951517833626238175733, 11.44896624550515594720021122212, 11.54785739355816888989907261646, 12.50358088755231677444729855335, 12.63004223667296844304306305156
