| L(s) = 1 | + 5.60·2-s − 1.89·3-s + 23.4·4-s − 5·5-s − 10.6·6-s + 11.4·7-s + 86.7·8-s − 23.4·9-s − 28.0·10-s + 37.7·11-s − 44.4·12-s − 8.69·13-s + 64.1·14-s + 9.47·15-s + 298.·16-s − 105.·17-s − 131.·18-s − 128.·19-s − 117.·20-s − 21.6·21-s + 211.·22-s − 23·23-s − 164.·24-s + 25·25-s − 48.7·26-s + 95.5·27-s + 268.·28-s + ⋯ |
| L(s) = 1 | + 1.98·2-s − 0.364·3-s + 2.93·4-s − 0.447·5-s − 0.723·6-s + 0.617·7-s + 3.83·8-s − 0.866·9-s − 0.886·10-s + 1.03·11-s − 1.06·12-s − 0.185·13-s + 1.22·14-s + 0.163·15-s + 4.66·16-s − 1.50·17-s − 1.71·18-s − 1.54·19-s − 1.31·20-s − 0.225·21-s + 2.05·22-s − 0.208·23-s − 1.39·24-s + 0.200·25-s − 0.367·26-s + 0.680·27-s + 1.81·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.387217660\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.387217660\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 5T \) |
| 23 | \( 1 + 23T \) |
| good | 2 | \( 1 - 5.60T + 8T^{2} \) |
| 3 | \( 1 + 1.89T + 27T^{2} \) |
| 7 | \( 1 - 11.4T + 343T^{2} \) |
| 11 | \( 1 - 37.7T + 1.33e3T^{2} \) |
| 13 | \( 1 + 8.69T + 2.19e3T^{2} \) |
| 17 | \( 1 + 105.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 128.T + 6.85e3T^{2} \) |
| 29 | \( 1 + 133.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 106.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 248.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 134.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 108.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 76.2T + 1.03e5T^{2} \) |
| 53 | \( 1 - 476.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 607.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 366.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 136.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 152.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.22e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 364.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 762.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 271.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 574.T + 9.12e5T^{2} \) |
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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
−13.10606026286804760985171368023, −12.09797136599461961746010360503, −11.38293641615412936303278623971, −10.77847380511645222129874223626, −8.462236193289258324322650966333, −6.94472487130905508025393066757, −6.08800160953761094023893853908, −4.80359286017242429493817590716, −3.88885640964164232945636930315, −2.19895551819804108901185471354,
2.19895551819804108901185471354, 3.88885640964164232945636930315, 4.80359286017242429493817590716, 6.08800160953761094023893853908, 6.94472487130905508025393066757, 8.462236193289258324322650966333, 10.77847380511645222129874223626, 11.38293641615412936303278623971, 12.09797136599461961746010360503, 13.10606026286804760985171368023
