| L(s) = 1 | + 4.21·2-s + 9.78·4-s + 11.4·5-s − 2.39·7-s + 7.51·8-s + 48.4·10-s − 63.3·11-s − 18.2·13-s − 10.0·14-s − 46.5·16-s − 14.9·17-s + 94.7·19-s + 112.·20-s − 267.·22-s + 2.03·23-s + 7.14·25-s − 76.9·26-s − 23.3·28-s + 86.4·29-s − 250.·31-s − 256.·32-s − 63.2·34-s − 27.4·35-s + 379.·37-s + 399.·38-s + 86.3·40-s + 125.·41-s + ⋯ |
| L(s) = 1 | + 1.49·2-s + 1.22·4-s + 1.02·5-s − 0.129·7-s + 0.332·8-s + 1.53·10-s − 1.73·11-s − 0.389·13-s − 0.192·14-s − 0.727·16-s − 0.213·17-s + 1.14·19-s + 1.25·20-s − 2.58·22-s + 0.0184·23-s + 0.0571·25-s − 0.580·26-s − 0.157·28-s + 0.553·29-s − 1.45·31-s − 1.41·32-s − 0.318·34-s − 0.132·35-s + 1.68·37-s + 1.70·38-s + 0.341·40-s + 0.476·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1899 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1899 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 211 | \( 1 + 211T \) |
| good | 2 | \( 1 - 4.21T + 8T^{2} \) |
| 5 | \( 1 - 11.4T + 125T^{2} \) |
| 7 | \( 1 + 2.39T + 343T^{2} \) |
| 11 | \( 1 + 63.3T + 1.33e3T^{2} \) |
| 13 | \( 1 + 18.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + 14.9T + 4.91e3T^{2} \) |
| 19 | \( 1 - 94.7T + 6.85e3T^{2} \) |
| 23 | \( 1 - 2.03T + 1.21e4T^{2} \) |
| 29 | \( 1 - 86.4T + 2.43e4T^{2} \) |
| 31 | \( 1 + 250.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 379.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 125.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 525.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 434.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 669.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 402.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 375.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 438.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 762.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 334.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.02e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 443.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 524.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 866.T + 9.12e5T^{2} \) |
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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
−8.308330591136465310591573336915, −7.46282231149589745925549144304, −6.54957926891125089616666998606, −5.75666840620132171337593331849, −5.21468452267847698214060142850, −4.63095814656964545069412796648, −3.30666978341072883689431665194, −2.70601992611824444595048168100, −1.79633995228675440294296241733, 0,
1.79633995228675440294296241733, 2.70601992611824444595048168100, 3.30666978341072883689431665194, 4.63095814656964545069412796648, 5.21468452267847698214060142850, 5.75666840620132171337593331849, 6.54957926891125089616666998606, 7.46282231149589745925549144304, 8.308330591136465310591573336915
