| L(s) = 1 | + 2.16·2-s + 2.66·4-s + 3.16·7-s + 1.44·8-s − 1.53·11-s + 5.24·13-s + 6.82·14-s − 2.21·16-s + 1.29·17-s + 5.44·19-s − 3.32·22-s − 6.44·23-s + 11.3·26-s + 8.43·28-s + 2.36·29-s − 4.46·31-s − 7.67·32-s + 2.78·34-s + 5.95·37-s + 11.7·38-s + 8.53·41-s + 8.48·43-s − 4.10·44-s − 13.9·46-s + 0.753·47-s + 2.98·49-s + 13.9·52-s + ⋯ |
| L(s) = 1 | + 1.52·2-s + 1.33·4-s + 1.19·7-s + 0.510·8-s − 0.463·11-s + 1.45·13-s + 1.82·14-s − 0.554·16-s + 0.313·17-s + 1.24·19-s − 0.708·22-s − 1.34·23-s + 2.22·26-s + 1.59·28-s + 0.438·29-s − 0.802·31-s − 1.35·32-s + 0.478·34-s + 0.979·37-s + 1.90·38-s + 1.33·41-s + 1.29·43-s − 0.618·44-s − 2.05·46-s + 0.109·47-s + 0.427·49-s + 1.93·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.906303604\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.906303604\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 2.16T + 2T^{2} \) |
| 7 | \( 1 - 3.16T + 7T^{2} \) |
| 11 | \( 1 + 1.53T + 11T^{2} \) |
| 13 | \( 1 - 5.24T + 13T^{2} \) |
| 17 | \( 1 - 1.29T + 17T^{2} \) |
| 19 | \( 1 - 5.44T + 19T^{2} \) |
| 23 | \( 1 + 6.44T + 23T^{2} \) |
| 29 | \( 1 - 2.36T + 29T^{2} \) |
| 31 | \( 1 + 4.46T + 31T^{2} \) |
| 37 | \( 1 - 5.95T + 37T^{2} \) |
| 41 | \( 1 - 8.53T + 41T^{2} \) |
| 43 | \( 1 - 8.48T + 43T^{2} \) |
| 47 | \( 1 - 0.753T + 47T^{2} \) |
| 53 | \( 1 - 9.74T + 53T^{2} \) |
| 59 | \( 1 + 4.11T + 59T^{2} \) |
| 61 | \( 1 + 10.6T + 61T^{2} \) |
| 67 | \( 1 - 1.89T + 67T^{2} \) |
| 71 | \( 1 - 0.0708T + 71T^{2} \) |
| 73 | \( 1 + 4.01T + 73T^{2} \) |
| 79 | \( 1 - 1.61T + 79T^{2} \) |
| 83 | \( 1 - 13.1T + 83T^{2} \) |
| 89 | \( 1 + 7.27T + 89T^{2} \) |
| 97 | \( 1 + 10.1T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.83072771642655105534557152565, −7.49472498164370808555050933047, −6.31312561268817389895538259222, −5.77523550518377746598562447170, −5.28363829110024112362568220935, −4.39349382703147158969647520908, −3.91366331788131046016456354563, −3.03754414463849198759619151508, −2.14207255947012100854716914697, −1.11108895344512822448540520459,
1.11108895344512822448540520459, 2.14207255947012100854716914697, 3.03754414463849198759619151508, 3.91366331788131046016456354563, 4.39349382703147158969647520908, 5.28363829110024112362568220935, 5.77523550518377746598562447170, 6.31312561268817389895538259222, 7.49472498164370808555050933047, 7.83072771642655105534557152565
