L(s) = 1 | + 2.12·2-s + 2.51·4-s − 0.484·5-s + 7-s + 1.09·8-s − 1.03·10-s + 5.60·13-s + 2.12·14-s − 2.70·16-s − 5.60·17-s − 5.28·19-s − 1.21·20-s − 2.48·23-s − 4.76·25-s + 11.9·26-s + 2.51·28-s − 5.28·29-s − 7.12·31-s − 7.93·32-s − 11.9·34-s − 0.484·35-s − 0.235·37-s − 11.2·38-s − 0.530·40-s − 2.39·41-s + 1.03·43-s − 5.28·46-s + ⋯ |
L(s) = 1 | + 1.50·2-s + 1.25·4-s − 0.216·5-s + 0.377·7-s + 0.387·8-s − 0.325·10-s + 1.55·13-s + 0.567·14-s − 0.676·16-s − 1.36·17-s − 1.21·19-s − 0.272·20-s − 0.518·23-s − 0.952·25-s + 2.33·26-s + 0.475·28-s − 0.980·29-s − 1.27·31-s − 1.40·32-s − 2.04·34-s − 0.0819·35-s − 0.0386·37-s − 1.82·38-s − 0.0839·40-s − 0.373·41-s + 0.157·43-s − 0.778·46-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
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 - 2.12T + 2T^{2} \) |
| 5 | \( 1 + 0.484T + 5T^{2} \) |
| 13 | \( 1 - 5.60T + 13T^{2} \) |
| 17 | \( 1 + 5.60T + 17T^{2} \) |
| 19 | \( 1 + 5.28T + 19T^{2} \) |
| 23 | \( 1 + 2.48T + 23T^{2} \) |
| 29 | \( 1 + 5.28T + 29T^{2} \) |
| 31 | \( 1 + 7.12T + 31T^{2} \) |
| 37 | \( 1 + 0.235T + 37T^{2} \) |
| 41 | \( 1 + 2.39T + 41T^{2} \) |
| 43 | \( 1 - 1.03T + 43T^{2} \) |
| 47 | \( 1 - 1.60T + 47T^{2} \) |
| 53 | \( 1 - 3.03T + 53T^{2} \) |
| 59 | \( 1 + 3.12T + 59T^{2} \) |
| 61 | \( 1 + 2.39T + 61T^{2} \) |
| 67 | \( 1 - 10.0T + 67T^{2} \) |
| 71 | \( 1 + 12.0T + 71T^{2} \) |
| 73 | \( 1 - 2.39T + 73T^{2} \) |
| 79 | \( 1 - 9.03T + 79T^{2} \) |
| 83 | \( 1 - 3.21T + 83T^{2} \) |
| 89 | \( 1 - 1.26T + 89T^{2} \) |
| 97 | \( 1 + 8.79T + 97T^{2} \) |
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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
−7.29609222558563396266763543826, −6.51237889931001551178237694924, −6.03648302897845718053503583793, −5.41422456700486232422723491694, −4.52718885811727722323314499442, −3.93486308076744415584564911596, −3.57297981054057704622203547003, −2.34208739529183519494339866583, −1.74635018172508328530954454478, 0,
1.74635018172508328530954454478, 2.34208739529183519494339866583, 3.57297981054057704622203547003, 3.93486308076744415584564911596, 4.52718885811727722323314499442, 5.41422456700486232422723491694, 6.03648302897845718053503583793, 6.51237889931001551178237694924, 7.29609222558563396266763543826