| L(s) = 1 | + 0.301·2-s − 1.90·4-s − 1.30·5-s − 7-s − 1.17·8-s − 0.391·10-s − 3.38·13-s − 0.301·14-s + 3.46·16-s − 0.478·17-s + 1.60·19-s + 2.48·20-s − 3.47·23-s − 3.30·25-s − 1.02·26-s + 1.90·28-s − 0.942·29-s − 1.17·31-s + 3.39·32-s − 0.144·34-s + 1.30·35-s − 3.55·37-s + 0.484·38-s + 1.53·40-s − 3.94·41-s − 2.87·43-s − 1.04·46-s + ⋯ |
| L(s) = 1 | + 0.212·2-s − 0.954·4-s − 0.581·5-s − 0.377·7-s − 0.416·8-s − 0.123·10-s − 0.939·13-s − 0.0804·14-s + 0.866·16-s − 0.116·17-s + 0.368·19-s + 0.555·20-s − 0.725·23-s − 0.661·25-s − 0.200·26-s + 0.360·28-s − 0.175·29-s − 0.211·31-s + 0.600·32-s − 0.0247·34-s + 0.219·35-s − 0.584·37-s + 0.0785·38-s + 0.242·40-s − 0.616·41-s − 0.437·43-s − 0.154·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5486674578\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5486674578\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 0.301T + 2T^{2} \) |
| 5 | \( 1 + 1.30T + 5T^{2} \) |
| 13 | \( 1 + 3.38T + 13T^{2} \) |
| 17 | \( 1 + 0.478T + 17T^{2} \) |
| 19 | \( 1 - 1.60T + 19T^{2} \) |
| 23 | \( 1 + 3.47T + 23T^{2} \) |
| 29 | \( 1 + 0.942T + 29T^{2} \) |
| 31 | \( 1 + 1.17T + 31T^{2} \) |
| 37 | \( 1 + 3.55T + 37T^{2} \) |
| 41 | \( 1 + 3.94T + 41T^{2} \) |
| 43 | \( 1 + 2.87T + 43T^{2} \) |
| 47 | \( 1 + 12.1T + 47T^{2} \) |
| 53 | \( 1 + 5.28T + 53T^{2} \) |
| 59 | \( 1 - 9.69T + 59T^{2} \) |
| 61 | \( 1 - 3.00T + 61T^{2} \) |
| 67 | \( 1 - 7.74T + 67T^{2} \) |
| 71 | \( 1 - 8.46T + 71T^{2} \) |
| 73 | \( 1 + 13.3T + 73T^{2} \) |
| 79 | \( 1 + 8.86T + 79T^{2} \) |
| 83 | \( 1 - 11.4T + 83T^{2} \) |
| 89 | \( 1 + 7.54T + 89T^{2} \) |
| 97 | \( 1 + 1.71T + 97T^{2} \) |
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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
−8.052969337524731223519596934619, −7.20161080518597139228418258260, −6.52214801459975582637959536470, −5.56849521000129267876799655955, −5.06203427997415907779799764274, −4.25549216870891828786498681340, −3.66056891330454772585855315355, −2.93609902946943926167987702600, −1.76808187883311624707727331352, −0.35165767904842287183835154178,
0.35165767904842287183835154178, 1.76808187883311624707727331352, 2.93609902946943926167987702600, 3.66056891330454772585855315355, 4.25549216870891828786498681340, 5.06203427997415907779799764274, 5.56849521000129267876799655955, 6.52214801459975582637959536470, 7.20161080518597139228418258260, 8.052969337524731223519596934619
