| L(s) = 1 | − 2-s + 2.42·3-s + 4-s + 2.98·5-s − 2.42·6-s + 1.97·7-s − 8-s + 2.88·9-s − 2.98·10-s + 1.52·11-s + 2.42·12-s + 3.43·13-s − 1.97·14-s + 7.24·15-s + 16-s − 1.93·17-s − 2.88·18-s + 7.28·19-s + 2.98·20-s + 4.79·21-s − 1.52·22-s + 23-s − 2.42·24-s + 3.92·25-s − 3.43·26-s − 0.285·27-s + 1.97·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.40·3-s + 0.5·4-s + 1.33·5-s − 0.990·6-s + 0.746·7-s − 0.353·8-s + 0.960·9-s − 0.944·10-s + 0.458·11-s + 0.700·12-s + 0.951·13-s − 0.527·14-s + 1.87·15-s + 0.250·16-s − 0.470·17-s − 0.679·18-s + 1.67·19-s + 0.668·20-s + 1.04·21-s − 0.324·22-s + 0.208·23-s − 0.495·24-s + 0.785·25-s − 0.672·26-s − 0.0548·27-s + 0.373·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.002208245\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.002208245\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 131 | \( 1 + T \) |
| good | 3 | \( 1 - 2.42T + 3T^{2} \) |
| 5 | \( 1 - 2.98T + 5T^{2} \) |
| 7 | \( 1 - 1.97T + 7T^{2} \) |
| 11 | \( 1 - 1.52T + 11T^{2} \) |
| 13 | \( 1 - 3.43T + 13T^{2} \) |
| 17 | \( 1 + 1.93T + 17T^{2} \) |
| 19 | \( 1 - 7.28T + 19T^{2} \) |
| 29 | \( 1 + 6.03T + 29T^{2} \) |
| 31 | \( 1 - 5.40T + 31T^{2} \) |
| 37 | \( 1 - 9.70T + 37T^{2} \) |
| 41 | \( 1 + 10.5T + 41T^{2} \) |
| 43 | \( 1 - 2.26T + 43T^{2} \) |
| 47 | \( 1 - 3.84T + 47T^{2} \) |
| 53 | \( 1 + 6.40T + 53T^{2} \) |
| 59 | \( 1 + 10.6T + 59T^{2} \) |
| 61 | \( 1 - 7.98T + 61T^{2} \) |
| 67 | \( 1 + 10.6T + 67T^{2} \) |
| 71 | \( 1 + 8.18T + 71T^{2} \) |
| 73 | \( 1 - 9.37T + 73T^{2} \) |
| 79 | \( 1 + 2.08T + 79T^{2} \) |
| 83 | \( 1 + 4.43T + 83T^{2} \) |
| 89 | \( 1 + 8.14T + 89T^{2} \) |
| 97 | \( 1 - 4.94T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.166183944999248936772903090267, −7.67318660679184668984543683314, −6.82309005204660227382220214078, −6.04136340182061480951375251776, −5.36296072822979096513706751677, −4.29924932803493571370550508057, −3.29384779295242757730887058206, −2.64761368302951786291314035187, −1.71390538696305694900037885877, −1.28749007518453348697644904383,
1.28749007518453348697644904383, 1.71390538696305694900037885877, 2.64761368302951786291314035187, 3.29384779295242757730887058206, 4.29924932803493571370550508057, 5.36296072822979096513706751677, 6.04136340182061480951375251776, 6.82309005204660227382220214078, 7.67318660679184668984543683314, 8.166183944999248936772903090267
