L(s) = 1 | − 2.71·2-s + 3-s + 5.38·4-s + 5-s − 2.71·6-s − 4.08·7-s − 9.20·8-s + 9-s − 2.71·10-s + 6.35·11-s + 5.38·12-s + 2.22·13-s + 11.1·14-s + 15-s + 14.2·16-s + 4.29·17-s − 2.71·18-s + 0.688·19-s + 5.38·20-s − 4.08·21-s − 17.2·22-s − 3.27·23-s − 9.20·24-s + 25-s − 6.06·26-s + 27-s − 22.0·28-s + ⋯ |
L(s) = 1 | − 1.92·2-s + 0.577·3-s + 2.69·4-s + 0.447·5-s − 1.10·6-s − 1.54·7-s − 3.25·8-s + 0.333·9-s − 0.859·10-s + 1.91·11-s + 1.55·12-s + 0.618·13-s + 2.96·14-s + 0.258·15-s + 3.56·16-s + 1.04·17-s − 0.640·18-s + 0.157·19-s + 1.20·20-s − 0.892·21-s − 3.68·22-s − 0.683·23-s − 1.87·24-s + 0.200·25-s − 1.18·26-s + 0.192·27-s − 4.16·28-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
\(L(1)\) |
\(\approx\) |
\(1.177955803\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.177955803\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.71T + 2T^{2} \) |
| 7 | \( 1 + 4.08T + 7T^{2} \) |
| 11 | \( 1 - 6.35T + 11T^{2} \) |
| 13 | \( 1 - 2.22T + 13T^{2} \) |
| 17 | \( 1 - 4.29T + 17T^{2} \) |
| 19 | \( 1 - 0.688T + 19T^{2} \) |
| 23 | \( 1 + 3.27T + 23T^{2} \) |
| 29 | \( 1 - 2.63T + 29T^{2} \) |
| 31 | \( 1 + 5.36T + 31T^{2} \) |
| 37 | \( 1 - 9.23T + 37T^{2} \) |
| 41 | \( 1 - 6.12T + 41T^{2} \) |
| 43 | \( 1 - 12.6T + 43T^{2} \) |
| 47 | \( 1 + 7.98T + 47T^{2} \) |
| 53 | \( 1 + 8.98T + 53T^{2} \) |
| 59 | \( 1 - 3.96T + 59T^{2} \) |
| 61 | \( 1 - 2.15T + 61T^{2} \) |
| 67 | \( 1 - 2.20T + 67T^{2} \) |
| 71 | \( 1 + 11.6T + 71T^{2} \) |
| 73 | \( 1 - 7.25T + 73T^{2} \) |
| 79 | \( 1 - 6.52T + 79T^{2} \) |
| 83 | \( 1 + 10.1T + 83T^{2} \) |
| 89 | \( 1 - 15.1T + 89T^{2} \) |
| 97 | \( 1 + 12.1T + 97T^{2} \) |
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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
−8.225323566839940741768492232999, −7.52878615067600969575242134520, −6.83192103534432075333822987083, −6.19191338222261793007217293886, −5.93930190612148507929796637630, −3.97680172434410343890152479680, −3.32347965197962771270312952199, −2.54811957443589925558802530220, −1.50485671785117711974632530745, −0.798684957922173078213712344169,
0.798684957922173078213712344169, 1.50485671785117711974632530745, 2.54811957443589925558802530220, 3.32347965197962771270312952199, 3.97680172434410343890152479680, 5.93930190612148507929796637630, 6.19191338222261793007217293886, 6.83192103534432075333822987083, 7.52878615067600969575242134520, 8.225323566839940741768492232999