| L(s) = 1 | − 0.156·2-s + 2.56·3-s − 1.97·4-s − 0.401·6-s + 1.09·7-s + 0.623·8-s + 3.57·9-s − 4.40·11-s − 5.06·12-s + 3.97·13-s − 0.171·14-s + 3.85·16-s + 6.22·17-s − 0.560·18-s + 6.97·19-s + 2.80·21-s + 0.690·22-s − 0.780·23-s + 1.59·24-s − 0.622·26-s + 1.47·27-s − 2.16·28-s − 29-s + 6.40·31-s − 1.85·32-s − 11.2·33-s − 0.975·34-s + ⋯ |
| L(s) = 1 | − 0.110·2-s + 1.48·3-s − 0.987·4-s − 0.164·6-s + 0.413·7-s + 0.220·8-s + 1.19·9-s − 1.32·11-s − 1.46·12-s + 1.10·13-s − 0.0458·14-s + 0.963·16-s + 1.50·17-s − 0.132·18-s + 1.60·19-s + 0.611·21-s + 0.147·22-s − 0.162·23-s + 0.326·24-s − 0.122·26-s + 0.282·27-s − 0.408·28-s − 0.185·29-s + 1.14·31-s − 0.327·32-s − 1.96·33-s − 0.167·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 725 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 725 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.046137393\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.046137393\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 + 0.156T + 2T^{2} \) |
| 3 | \( 1 - 2.56T + 3T^{2} \) |
| 7 | \( 1 - 1.09T + 7T^{2} \) |
| 11 | \( 1 + 4.40T + 11T^{2} \) |
| 13 | \( 1 - 3.97T + 13T^{2} \) |
| 17 | \( 1 - 6.22T + 17T^{2} \) |
| 19 | \( 1 - 6.97T + 19T^{2} \) |
| 23 | \( 1 + 0.780T + 23T^{2} \) |
| 31 | \( 1 - 6.40T + 31T^{2} \) |
| 37 | \( 1 + 1.09T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 0.376T + 43T^{2} \) |
| 47 | \( 1 + 4.75T + 47T^{2} \) |
| 53 | \( 1 + 11.2T + 53T^{2} \) |
| 59 | \( 1 - 10T + 59T^{2} \) |
| 61 | \( 1 + 1.14T + 61T^{2} \) |
| 67 | \( 1 + 5.90T + 67T^{2} \) |
| 71 | \( 1 - 2T + 71T^{2} \) |
| 73 | \( 1 - 8.72T + 73T^{2} \) |
| 79 | \( 1 + 5.54T + 79T^{2} \) |
| 83 | \( 1 - 6.22T + 83T^{2} \) |
| 89 | \( 1 + 10.8T + 89T^{2} \) |
| 97 | \( 1 + 17.4T + 97T^{2} \) |
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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
−9.895311590222757810429675142167, −9.639667438170062703484818102138, −8.396885172015545131343907801498, −8.149904819324278150609699915443, −7.45497048722365686259751622533, −5.71190249399325740557940185203, −4.87166191367786617660685042827, −3.60633228716475891852470462727, −2.97379630732172623355533401360, −1.31947208772224151974113011348,
1.31947208772224151974113011348, 2.97379630732172623355533401360, 3.60633228716475891852470462727, 4.87166191367786617660685042827, 5.71190249399325740557940185203, 7.45497048722365686259751622533, 8.149904819324278150609699915443, 8.396885172015545131343907801498, 9.639667438170062703484818102138, 9.895311590222757810429675142167
