| L(s) = 1 | − 2-s + 4-s + 2·5-s + 7-s − 8-s − 2·10-s + 4.82·11-s + 5.65·13-s − 14-s + 16-s + 6.82·17-s − 19-s + 2·20-s − 4.82·22-s + 4·23-s − 25-s − 5.65·26-s + 28-s + 2·29-s − 4.82·31-s − 32-s − 6.82·34-s + 2·35-s − 8.48·37-s + 38-s − 2·40-s − 3.65·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 0.894·5-s + 0.377·7-s − 0.353·8-s − 0.632·10-s + 1.45·11-s + 1.56·13-s − 0.267·14-s + 0.250·16-s + 1.65·17-s − 0.229·19-s + 0.447·20-s − 1.02·22-s + 0.834·23-s − 0.200·25-s − 1.10·26-s + 0.188·28-s + 0.371·29-s − 0.867·31-s − 0.176·32-s − 1.17·34-s + 0.338·35-s − 1.39·37-s + 0.162·38-s − 0.316·40-s − 0.571·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.053449934\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.053449934\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 - 2T + 5T^{2} \) |
| 11 | \( 1 - 4.82T + 11T^{2} \) |
| 13 | \( 1 - 5.65T + 13T^{2} \) |
| 17 | \( 1 - 6.82T + 17T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 + 4.82T + 31T^{2} \) |
| 37 | \( 1 + 8.48T + 37T^{2} \) |
| 41 | \( 1 + 3.65T + 41T^{2} \) |
| 43 | \( 1 - 9.65T + 43T^{2} \) |
| 47 | \( 1 + 4.48T + 47T^{2} \) |
| 53 | \( 1 + 7.65T + 53T^{2} \) |
| 59 | \( 1 - 8T + 59T^{2} \) |
| 61 | \( 1 + 13.3T + 61T^{2} \) |
| 67 | \( 1 + 6.82T + 67T^{2} \) |
| 71 | \( 1 + 5.65T + 71T^{2} \) |
| 73 | \( 1 + 7.65T + 73T^{2} \) |
| 79 | \( 1 - 7.65T + 79T^{2} \) |
| 83 | \( 1 - 11.6T + 83T^{2} \) |
| 89 | \( 1 - 13.3T + 89T^{2} \) |
| 97 | \( 1 + 18.4T + 97T^{2} \) |
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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
−9.091235061615110192500441976884, −8.367850581834467909623872376838, −7.51564378683913989797805091718, −6.58411837978721256143834293048, −6.01953831112159033340689169029, −5.27580575356770397780687947488, −3.93433865226781707539049283785, −3.15970766137001751001675268124, −1.66239463988881447462814017537, −1.21822951447109040137952411219,
1.21822951447109040137952411219, 1.66239463988881447462814017537, 3.15970766137001751001675268124, 3.93433865226781707539049283785, 5.27580575356770397780687947488, 6.01953831112159033340689169029, 6.58411837978721256143834293048, 7.51564378683913989797805091718, 8.367850581834467909623872376838, 9.091235061615110192500441976884
