| L(s) = 1 | + 4.54·2-s + 12.6·4-s + 12.9·5-s − 16.7·7-s + 21.2·8-s + 58.7·10-s + 24.9·11-s − 76.0·14-s − 4.67·16-s + 134.·17-s + 14.9·19-s + 163.·20-s + 113.·22-s − 72·23-s + 41.7·25-s − 212.·28-s + 206.·29-s + 249.·31-s − 191.·32-s + 610.·34-s − 216·35-s − 293.·37-s + 67.9·38-s + 274.·40-s + 250.·41-s + 432.·43-s + 316.·44-s + ⋯ |
| L(s) = 1 | + 1.60·2-s + 1.58·4-s + 1.15·5-s − 0.903·7-s + 0.940·8-s + 1.85·10-s + 0.683·11-s − 1.45·14-s − 0.0731·16-s + 1.91·17-s + 0.180·19-s + 1.83·20-s + 1.09·22-s − 0.652·23-s + 0.333·25-s − 1.43·28-s + 1.32·29-s + 1.44·31-s − 1.05·32-s + 3.07·34-s − 1.04·35-s − 1.30·37-s + 0.289·38-s + 1.08·40-s + 0.954·41-s + 1.53·43-s + 1.08·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(7.283995060\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.283995060\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 4.54T + 8T^{2} \) |
| 5 | \( 1 - 12.9T + 125T^{2} \) |
| 7 | \( 1 + 16.7T + 343T^{2} \) |
| 11 | \( 1 - 24.9T + 1.33e3T^{2} \) |
| 17 | \( 1 - 134.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 14.9T + 6.85e3T^{2} \) |
| 23 | \( 1 + 72T + 1.21e4T^{2} \) |
| 29 | \( 1 - 206.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 249.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 293.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 250.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 432.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 159.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 194.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 232.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 185.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 39.4T + 3.00e5T^{2} \) |
| 71 | \( 1 - 920.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 549.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 933.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.09e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 532.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 362.T + 9.12e5T^{2} \) |
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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
−9.428833966910404591842866887695, −8.188657192235096697212229769006, −6.99533031557728530491778567558, −6.27761059688710239072835316572, −5.82243433727049161973506361052, −5.05595570024576076584647712644, −3.97478438547376408660822377134, −3.18473139534580197253669217107, −2.37907505748095102302664250422, −1.09659231418290510296333234175,
1.09659231418290510296333234175, 2.37907505748095102302664250422, 3.18473139534580197253669217107, 3.97478438547376408660822377134, 5.05595570024576076584647712644, 5.82243433727049161973506361052, 6.27761059688710239072835316572, 6.99533031557728530491778567558, 8.188657192235096697212229769006, 9.428833966910404591842866887695
