| L(s) = 1 | + 2.20·2-s + 3-s + 2.87·4-s − 5-s + 2.20·6-s − 4.54·7-s + 1.92·8-s + 9-s − 2.20·10-s + 2.55·11-s + 2.87·12-s + 3.87·13-s − 10.0·14-s − 15-s − 1.49·16-s + 1.55·17-s + 2.20·18-s + 3.64·19-s − 2.87·20-s − 4.54·21-s + 5.63·22-s − 3.24·23-s + 1.92·24-s + 25-s + 8.55·26-s + 27-s − 13.0·28-s + ⋯ |
| L(s) = 1 | + 1.56·2-s + 0.577·3-s + 1.43·4-s − 0.447·5-s + 0.901·6-s − 1.71·7-s + 0.679·8-s + 0.333·9-s − 0.697·10-s + 0.770·11-s + 0.828·12-s + 1.07·13-s − 2.68·14-s − 0.258·15-s − 0.374·16-s + 0.376·17-s + 0.520·18-s + 0.836·19-s − 0.642·20-s − 0.992·21-s + 1.20·22-s − 0.676·23-s + 0.392·24-s + 0.200·25-s + 1.67·26-s + 0.192·27-s − 2.46·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.134733508\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.134733508\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 401 | \( 1 + T \) |
| good | 2 | \( 1 - 2.20T + 2T^{2} \) |
| 7 | \( 1 + 4.54T + 7T^{2} \) |
| 11 | \( 1 - 2.55T + 11T^{2} \) |
| 13 | \( 1 - 3.87T + 13T^{2} \) |
| 17 | \( 1 - 1.55T + 17T^{2} \) |
| 19 | \( 1 - 3.64T + 19T^{2} \) |
| 23 | \( 1 + 3.24T + 23T^{2} \) |
| 29 | \( 1 - 1.54T + 29T^{2} \) |
| 31 | \( 1 - 7.49T + 31T^{2} \) |
| 37 | \( 1 - 4.52T + 37T^{2} \) |
| 41 | \( 1 + 0.126T + 41T^{2} \) |
| 43 | \( 1 + 1.83T + 43T^{2} \) |
| 47 | \( 1 - 3.98T + 47T^{2} \) |
| 53 | \( 1 - 8.31T + 53T^{2} \) |
| 59 | \( 1 + 6.81T + 59T^{2} \) |
| 61 | \( 1 + 7.17T + 61T^{2} \) |
| 67 | \( 1 - 14.3T + 67T^{2} \) |
| 71 | \( 1 - 10.1T + 71T^{2} \) |
| 73 | \( 1 - 11.7T + 73T^{2} \) |
| 79 | \( 1 - 3.41T + 79T^{2} \) |
| 83 | \( 1 + 7.55T + 83T^{2} \) |
| 89 | \( 1 - 15.3T + 89T^{2} \) |
| 97 | \( 1 - 12.5T + 97T^{2} \) |
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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
−7.930361677165046493427423657195, −7.05581820630929047532617370065, −6.35858551447012925433079492423, −6.12451074460801115256457838215, −5.11691133224498515121495008088, −4.10080536873863172532634651768, −3.68201385050828070252977142218, −3.18082673178847098525473609141, −2.41712683719111114348181647832, −0.923617277100366107209873319607,
0.923617277100366107209873319607, 2.41712683719111114348181647832, 3.18082673178847098525473609141, 3.68201385050828070252977142218, 4.10080536873863172532634651768, 5.11691133224498515121495008088, 6.12451074460801115256457838215, 6.35858551447012925433079492423, 7.05581820630929047532617370065, 7.930361677165046493427423657195
