| L(s) = 1 | − 1.67·2-s − 2.69·3-s + 0.817·4-s + 4.51·6-s + 2.51·7-s + 1.98·8-s + 4.24·9-s − 11-s − 2.20·12-s + 3.42·13-s − 4.22·14-s − 4.96·16-s + 1.55·17-s − 7.12·18-s − 19-s − 6.76·21-s + 1.67·22-s + 8.59·23-s − 5.34·24-s − 5.74·26-s − 3.34·27-s + 2.05·28-s − 0.732·29-s + 9.53·31-s + 4.36·32-s + 2.69·33-s − 2.61·34-s + ⋯ |
| L(s) = 1 | − 1.18·2-s − 1.55·3-s + 0.408·4-s + 1.84·6-s + 0.950·7-s + 0.701·8-s + 1.41·9-s − 0.301·11-s − 0.635·12-s + 0.949·13-s − 1.12·14-s − 1.24·16-s + 0.378·17-s − 1.67·18-s − 0.229·19-s − 1.47·21-s + 0.357·22-s + 1.79·23-s − 1.09·24-s − 1.12·26-s − 0.643·27-s + 0.388·28-s − 0.135·29-s + 1.71·31-s + 0.772·32-s + 0.468·33-s − 0.448·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7869528681\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7869528681\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 + 1.67T + 2T^{2} \) |
| 3 | \( 1 + 2.69T + 3T^{2} \) |
| 7 | \( 1 - 2.51T + 7T^{2} \) |
| 13 | \( 1 - 3.42T + 13T^{2} \) |
| 17 | \( 1 - 1.55T + 17T^{2} \) |
| 23 | \( 1 - 8.59T + 23T^{2} \) |
| 29 | \( 1 + 0.732T + 29T^{2} \) |
| 31 | \( 1 - 9.53T + 31T^{2} \) |
| 37 | \( 1 - 7.74T + 37T^{2} \) |
| 41 | \( 1 - 3.46T + 41T^{2} \) |
| 43 | \( 1 + 5.41T + 43T^{2} \) |
| 47 | \( 1 - 12.0T + 47T^{2} \) |
| 53 | \( 1 - 5.37T + 53T^{2} \) |
| 59 | \( 1 - 7.67T + 59T^{2} \) |
| 61 | \( 1 - 10.7T + 61T^{2} \) |
| 67 | \( 1 - 3.64T + 67T^{2} \) |
| 71 | \( 1 - 5.47T + 71T^{2} \) |
| 73 | \( 1 - 8.79T + 73T^{2} \) |
| 79 | \( 1 + 2.28T + 79T^{2} \) |
| 83 | \( 1 + 6.13T + 83T^{2} \) |
| 89 | \( 1 + 11.0T + 89T^{2} \) |
| 97 | \( 1 - 16.8T + 97T^{2} \) |
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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
−8.320227882260400766433457795622, −7.53165531996730570432618247660, −6.85921884897379407497604570956, −6.13837445555726290626242306121, −5.26881883237955701503842383892, −4.79504860711191478653742613369, −3.98231818688839642234228524191, −2.45577272197231390594090037979, −1.13069542794891383401458382434, −0.823280102751421831213408256847,
0.823280102751421831213408256847, 1.13069542794891383401458382434, 2.45577272197231390594090037979, 3.98231818688839642234228524191, 4.79504860711191478653742613369, 5.26881883237955701503842383892, 6.13837445555726290626242306121, 6.85921884897379407497604570956, 7.53165531996730570432618247660, 8.320227882260400766433457795622
