| L(s) = 1 | + 47.3·2-s − 2.03e4·3-s − 1.28e5·4-s + 1.59e6·5-s − 9.61e5·6-s + 4.91e6·7-s − 1.23e7·8-s + 2.83e8·9-s + 7.53e7·10-s − 1.25e9·11-s + 2.61e9·12-s − 2.74e9·13-s + 2.32e8·14-s − 3.23e10·15-s + 1.63e10·16-s − 2.49e10·17-s + 1.34e10·18-s + 8.65e10·19-s − 2.05e11·20-s − 9.97e10·21-s − 5.92e10·22-s + 1.91e11·23-s + 2.49e11·24-s + 1.77e12·25-s − 1.29e11·26-s − 3.14e12·27-s − 6.32e11·28-s + ⋯ |
| L(s) = 1 | + 0.130·2-s − 1.78·3-s − 0.982·4-s + 1.82·5-s − 0.233·6-s + 0.321·7-s − 0.259·8-s + 2.19·9-s + 0.238·10-s − 1.75·11-s + 1.75·12-s − 0.932·13-s + 0.0420·14-s − 3.25·15-s + 0.949·16-s − 0.865·17-s + 0.287·18-s + 1.16·19-s − 1.79·20-s − 0.575·21-s − 0.230·22-s + 0.510·23-s + 0.463·24-s + 2.32·25-s − 0.121·26-s − 2.13·27-s − 0.316·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(9)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{19}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 59 | \( 1 + 1.46e14T \) |
| good | 2 | \( 1 - 47.3T + 1.31e5T^{2} \) |
| 3 | \( 1 + 2.03e4T + 1.29e8T^{2} \) |
| 5 | \( 1 - 1.59e6T + 7.62e11T^{2} \) |
| 7 | \( 1 - 4.91e6T + 2.32e14T^{2} \) |
| 11 | \( 1 + 1.25e9T + 5.05e17T^{2} \) |
| 13 | \( 1 + 2.74e9T + 8.65e18T^{2} \) |
| 17 | \( 1 + 2.49e10T + 8.27e20T^{2} \) |
| 19 | \( 1 - 8.65e10T + 5.48e21T^{2} \) |
| 23 | \( 1 - 1.91e11T + 1.41e23T^{2} \) |
| 29 | \( 1 + 3.66e12T + 7.25e24T^{2} \) |
| 31 | \( 1 - 5.11e12T + 2.25e25T^{2} \) |
| 37 | \( 1 + 2.34e13T + 4.56e26T^{2} \) |
| 41 | \( 1 - 6.34e13T + 2.61e27T^{2} \) |
| 43 | \( 1 - 8.45e13T + 5.87e27T^{2} \) |
| 47 | \( 1 - 2.43e14T + 2.66e28T^{2} \) |
| 53 | \( 1 - 3.48e14T + 2.05e29T^{2} \) |
| 61 | \( 1 + 1.88e15T + 2.24e30T^{2} \) |
| 67 | \( 1 + 4.55e14T + 1.10e31T^{2} \) |
| 71 | \( 1 - 6.88e15T + 2.96e31T^{2} \) |
| 73 | \( 1 - 1.37e15T + 4.74e31T^{2} \) |
| 79 | \( 1 - 1.20e16T + 1.81e32T^{2} \) |
| 83 | \( 1 + 3.21e16T + 4.21e32T^{2} \) |
| 89 | \( 1 + 1.50e16T + 1.37e33T^{2} \) |
| 97 | \( 1 - 9.68e14T + 5.95e33T^{2} \) |
| 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
−10.89106229929556275459823804116, −10.14211357651262783710326151030, −9.312236991695971501009463344000, −7.37053845464158615811383814930, −5.90322897724668733026818333455, −5.29441333982968813125894476912, −4.76138226094413725417952835118, −2.42372148083583977988297220334, −1.04147141334174705078828472076, 0,
1.04147141334174705078828472076, 2.42372148083583977988297220334, 4.76138226094413725417952835118, 5.29441333982968813125894476912, 5.90322897724668733026818333455, 7.37053845464158615811383814930, 9.312236991695971501009463344000, 10.14211357651262783710326151030, 10.89106229929556275459823804116
