| L(s) = 1 | − 2.14·2-s + 2.59·4-s + 1.59·5-s − 3.96·7-s − 1.27·8-s − 3.42·10-s + 0.856·11-s + 2.22·13-s + 8.51·14-s − 2.45·16-s + 2.49·17-s + 19-s + 4.14·20-s − 1.83·22-s + 6.08·23-s − 2.45·25-s − 4.76·26-s − 10.3·28-s + 2.04·29-s − 1.77·31-s + 7.81·32-s − 5.34·34-s − 6.33·35-s − 4.42·37-s − 2.14·38-s − 2.04·40-s + 10.7·41-s + ⋯ |
| L(s) = 1 | − 1.51·2-s + 1.29·4-s + 0.713·5-s − 1.50·7-s − 0.451·8-s − 1.08·10-s + 0.258·11-s + 0.616·13-s + 2.27·14-s − 0.613·16-s + 0.604·17-s + 0.229·19-s + 0.926·20-s − 0.391·22-s + 1.26·23-s − 0.490·25-s − 0.934·26-s − 1.94·28-s + 0.380·29-s − 0.317·31-s + 1.38·32-s − 0.916·34-s − 1.07·35-s − 0.726·37-s − 0.347·38-s − 0.322·40-s + 1.68·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8037 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8037 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8900393621\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8900393621\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 19 | \( 1 - T \) |
| 47 | \( 1 - T \) |
| good | 2 | \( 1 + 2.14T + 2T^{2} \) |
| 5 | \( 1 - 1.59T + 5T^{2} \) |
| 7 | \( 1 + 3.96T + 7T^{2} \) |
| 11 | \( 1 - 0.856T + 11T^{2} \) |
| 13 | \( 1 - 2.22T + 13T^{2} \) |
| 17 | \( 1 - 2.49T + 17T^{2} \) |
| 23 | \( 1 - 6.08T + 23T^{2} \) |
| 29 | \( 1 - 2.04T + 29T^{2} \) |
| 31 | \( 1 + 1.77T + 31T^{2} \) |
| 37 | \( 1 + 4.42T + 37T^{2} \) |
| 41 | \( 1 - 10.7T + 41T^{2} \) |
| 43 | \( 1 - 8.40T + 43T^{2} \) |
| 53 | \( 1 + 11.9T + 53T^{2} \) |
| 59 | \( 1 - 7.93T + 59T^{2} \) |
| 61 | \( 1 - 4.55T + 61T^{2} \) |
| 67 | \( 1 + 5.15T + 67T^{2} \) |
| 71 | \( 1 + 12.7T + 71T^{2} \) |
| 73 | \( 1 + 0.0857T + 73T^{2} \) |
| 79 | \( 1 - 3.11T + 79T^{2} \) |
| 83 | \( 1 - 4.90T + 83T^{2} \) |
| 89 | \( 1 - 9.11T + 89T^{2} \) |
| 97 | \( 1 - 6.98T + 97T^{2} \) |
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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
−7.81370916882139840800993909639, −7.31058607770103772936585153642, −6.51665432329153677614443369817, −6.12441476484050215579768015812, −5.30750153556067499575989901663, −4.11965446100092550689732445523, −3.21289704420132975455129534493, −2.49641273750927279557430407531, −1.43630338395781584596234078352, −0.63029577899228774220481706800,
0.63029577899228774220481706800, 1.43630338395781584596234078352, 2.49641273750927279557430407531, 3.21289704420132975455129534493, 4.11965446100092550689732445523, 5.30750153556067499575989901663, 6.12441476484050215579768015812, 6.51665432329153677614443369817, 7.31058607770103772936585153642, 7.81370916882139840800993909639
