| L(s) = 1 | + 2-s + 1.47·3-s + 4-s − 3.42·5-s + 1.47·6-s − 2.00·7-s + 8-s − 0.836·9-s − 3.42·10-s + 3.77·11-s + 1.47·12-s − 4.09·13-s − 2.00·14-s − 5.03·15-s + 16-s − 2.88·17-s − 0.836·18-s + 1.26·19-s − 3.42·20-s − 2.94·21-s + 3.77·22-s − 9.00·23-s + 1.47·24-s + 6.72·25-s − 4.09·26-s − 5.64·27-s − 2.00·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.849·3-s + 0.5·4-s − 1.53·5-s + 0.600·6-s − 0.756·7-s + 0.353·8-s − 0.278·9-s − 1.08·10-s + 1.13·11-s + 0.424·12-s − 1.13·13-s − 0.535·14-s − 1.30·15-s + 0.250·16-s − 0.700·17-s − 0.197·18-s + 0.289·19-s − 0.765·20-s − 0.642·21-s + 0.805·22-s − 1.87·23-s + 0.300·24-s + 1.34·25-s − 0.802·26-s − 1.08·27-s − 0.378·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1502 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1502 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 751 | \( 1 - T \) |
| good | 3 | \( 1 - 1.47T + 3T^{2} \) |
| 5 | \( 1 + 3.42T + 5T^{2} \) |
| 7 | \( 1 + 2.00T + 7T^{2} \) |
| 11 | \( 1 - 3.77T + 11T^{2} \) |
| 13 | \( 1 + 4.09T + 13T^{2} \) |
| 17 | \( 1 + 2.88T + 17T^{2} \) |
| 19 | \( 1 - 1.26T + 19T^{2} \) |
| 23 | \( 1 + 9.00T + 23T^{2} \) |
| 29 | \( 1 - 0.872T + 29T^{2} \) |
| 31 | \( 1 + 5.05T + 31T^{2} \) |
| 37 | \( 1 + 4.53T + 37T^{2} \) |
| 41 | \( 1 + 4.02T + 41T^{2} \) |
| 43 | \( 1 - 10.6T + 43T^{2} \) |
| 47 | \( 1 - 6.17T + 47T^{2} \) |
| 53 | \( 1 + 6.04T + 53T^{2} \) |
| 59 | \( 1 - 4.45T + 59T^{2} \) |
| 61 | \( 1 + 12.5T + 61T^{2} \) |
| 67 | \( 1 - 6.45T + 67T^{2} \) |
| 71 | \( 1 - 10.5T + 71T^{2} \) |
| 73 | \( 1 + 8.35T + 73T^{2} \) |
| 79 | \( 1 + 3.70T + 79T^{2} \) |
| 83 | \( 1 - 3.33T + 83T^{2} \) |
| 89 | \( 1 - 0.0960T + 89T^{2} \) |
| 97 | \( 1 + 6.21T + 97T^{2} \) |
| 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
−9.034172111046878997167042382766, −8.144279480326944783864849918843, −7.47803123479985722237674811810, −6.77941109359910191408881165465, −5.80022907585256237399463055442, −4.48937867999717958709407535017, −3.84097841276622139976028614269, −3.22896194067637889980888364095, −2.15448331663338875077569414384, 0,
2.15448331663338875077569414384, 3.22896194067637889980888364095, 3.84097841276622139976028614269, 4.48937867999717958709407535017, 5.80022907585256237399463055442, 6.77941109359910191408881165465, 7.47803123479985722237674811810, 8.144279480326944783864849918843, 9.034172111046878997167042382766
