| L(s) = 1 | + 2.15·2-s + 6.64·3-s − 3.34·4-s − 5·5-s + 14.3·6-s + 2.49·7-s − 24.4·8-s + 17.1·9-s − 10.7·10-s + 3.28·11-s − 22.2·12-s − 13·13-s + 5.38·14-s − 33.2·15-s − 26.0·16-s + 57.2·17-s + 37.0·18-s + 75.7·19-s + 16.7·20-s + 16.5·21-s + 7.08·22-s + 48.5·23-s − 162.·24-s + 25·25-s − 28.0·26-s − 65.4·27-s − 8.34·28-s + ⋯ |
| L(s) = 1 | + 0.762·2-s + 1.27·3-s − 0.417·4-s − 0.447·5-s + 0.975·6-s + 0.134·7-s − 1.08·8-s + 0.635·9-s − 0.341·10-s + 0.0900·11-s − 0.534·12-s − 0.277·13-s + 0.102·14-s − 0.571·15-s − 0.407·16-s + 0.817·17-s + 0.484·18-s + 0.914·19-s + 0.186·20-s + 0.172·21-s + 0.0686·22-s + 0.439·23-s − 1.38·24-s + 0.200·25-s − 0.211·26-s − 0.466·27-s − 0.0562·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 5T \) |
| 13 | \( 1 + 13T \) |
| 31 | \( 1 - 31T \) |
| good | 2 | \( 1 - 2.15T + 8T^{2} \) |
| 3 | \( 1 - 6.64T + 27T^{2} \) |
| 7 | \( 1 - 2.49T + 343T^{2} \) |
| 11 | \( 1 - 3.28T + 1.33e3T^{2} \) |
| 17 | \( 1 - 57.2T + 4.91e3T^{2} \) |
| 19 | \( 1 - 75.7T + 6.85e3T^{2} \) |
| 23 | \( 1 - 48.5T + 1.21e4T^{2} \) |
| 29 | \( 1 + 188.T + 2.43e4T^{2} \) |
| 37 | \( 1 + 173.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 350.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 67.5T + 7.95e4T^{2} \) |
| 47 | \( 1 - 109.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 303.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 599.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 433.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 711.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 62.3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 22.9T + 3.89e5T^{2} \) |
| 79 | \( 1 + 8.19T + 4.93e5T^{2} \) |
| 83 | \( 1 + 445.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 367.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 879.T + 9.12e5T^{2} \) |
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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
−8.370905007304959479226946877316, −7.76889693509293814299453272862, −6.99568638283132147957347998393, −5.75037289985073845099296150819, −5.05427961319820539315838862254, −4.10206762818350806926364765828, −3.36009080386947440727205195047, −2.86841582397445372144182407593, −1.50765447609246586324240397202, 0,
1.50765447609246586324240397202, 2.86841582397445372144182407593, 3.36009080386947440727205195047, 4.10206762818350806926364765828, 5.05427961319820539315838862254, 5.75037289985073845099296150819, 6.99568638283132147957347998393, 7.76889693509293814299453272862, 8.370905007304959479226946877316
