| L(s) = 1 | − 1.92·2-s − 0.621·3-s + 1.70·4-s + 1.19·6-s − 5.15·7-s + 0.574·8-s − 2.61·9-s + 11-s − 1.05·12-s − 0.138·13-s + 9.91·14-s − 4.50·16-s − 1.86·17-s + 5.02·18-s + 19-s + 3.20·21-s − 1.92·22-s − 0.167·23-s − 0.356·24-s + 0.265·26-s + 3.48·27-s − 8.76·28-s + 4.80·29-s − 1.73·31-s + 7.52·32-s − 0.621·33-s + 3.58·34-s + ⋯ |
| L(s) = 1 | − 1.36·2-s − 0.358·3-s + 0.850·4-s + 0.487·6-s − 1.94·7-s + 0.203·8-s − 0.871·9-s + 0.301·11-s − 0.305·12-s − 0.0383·13-s + 2.64·14-s − 1.12·16-s − 0.451·17-s + 1.18·18-s + 0.229·19-s + 0.698·21-s − 0.410·22-s − 0.0349·23-s − 0.0728·24-s + 0.0521·26-s + 0.671·27-s − 1.65·28-s + 0.893·29-s − 0.310·31-s + 1.33·32-s − 0.108·33-s + 0.614·34-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.92T + 2T^{2} \) |
| 3 | \( 1 + 0.621T + 3T^{2} \) |
| 7 | \( 1 + 5.15T + 7T^{2} \) |
| 13 | \( 1 + 0.138T + 13T^{2} \) |
| 17 | \( 1 + 1.86T + 17T^{2} \) |
| 23 | \( 1 + 0.167T + 23T^{2} \) |
| 29 | \( 1 - 4.80T + 29T^{2} \) |
| 31 | \( 1 + 1.73T + 31T^{2} \) |
| 37 | \( 1 + 0.442T + 37T^{2} \) |
| 41 | \( 1 + 12.1T + 41T^{2} \) |
| 43 | \( 1 - 8.67T + 43T^{2} \) |
| 47 | \( 1 + 2.21T + 47T^{2} \) |
| 53 | \( 1 + 1.34T + 53T^{2} \) |
| 59 | \( 1 - 3.15T + 59T^{2} \) |
| 61 | \( 1 - 1.90T + 61T^{2} \) |
| 67 | \( 1 - 10.7T + 67T^{2} \) |
| 71 | \( 1 - 11.2T + 71T^{2} \) |
| 73 | \( 1 - 9.78T + 73T^{2} \) |
| 79 | \( 1 + 6.14T + 79T^{2} \) |
| 83 | \( 1 - 0.190T + 83T^{2} \) |
| 89 | \( 1 + 8.94T + 89T^{2} \) |
| 97 | \( 1 + 6.74T + 97T^{2} \) |
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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
−8.091915237425398658566712173399, −6.95835405075735052392808925420, −6.71966233594767634834889456165, −5.97307487730567693459398342733, −5.09015170252371211248867616346, −3.93491133697396810520223905644, −3.09309470386360516855148104359, −2.26078149197234511023587190563, −0.825998239999587926246552402833, 0,
0.825998239999587926246552402833, 2.26078149197234511023587190563, 3.09309470386360516855148104359, 3.93491133697396810520223905644, 5.09015170252371211248867616346, 5.97307487730567693459398342733, 6.71966233594767634834889456165, 6.95835405075735052392808925420, 8.091915237425398658566712173399
