| L(s) = 1 | + 3.85·2-s + 6.82·4-s + 5·5-s − 26.3·7-s − 4.52·8-s + 19.2·10-s + 5.07·11-s − 82.7·13-s − 101.·14-s − 72.0·16-s − 52.5·17-s − 29.8·19-s + 34.1·20-s + 19.5·22-s + 98.2·23-s + 25·25-s − 318.·26-s − 179.·28-s + 167.·29-s + 190.·31-s − 241.·32-s − 202.·34-s − 131.·35-s − 365.·37-s − 114.·38-s − 22.6·40-s − 111.·41-s + ⋯ |
| L(s) = 1 | + 1.36·2-s + 0.853·4-s + 0.447·5-s − 1.42·7-s − 0.199·8-s + 0.608·10-s + 0.139·11-s − 1.76·13-s − 1.93·14-s − 1.12·16-s − 0.750·17-s − 0.360·19-s + 0.381·20-s + 0.189·22-s + 0.890·23-s + 0.200·25-s − 2.40·26-s − 1.21·28-s + 1.07·29-s + 1.10·31-s − 1.33·32-s − 1.02·34-s − 0.635·35-s − 1.62·37-s − 0.490·38-s − 0.0894·40-s − 0.425·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| good | 2 | \( 1 - 3.85T + 8T^{2} \) |
| 7 | \( 1 + 26.3T + 343T^{2} \) |
| 11 | \( 1 - 5.07T + 1.33e3T^{2} \) |
| 13 | \( 1 + 82.7T + 2.19e3T^{2} \) |
| 17 | \( 1 + 52.5T + 4.91e3T^{2} \) |
| 19 | \( 1 + 29.8T + 6.85e3T^{2} \) |
| 23 | \( 1 - 98.2T + 1.21e4T^{2} \) |
| 29 | \( 1 - 167.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 190.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 365.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 111.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 403.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 232.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 410.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 152.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 532.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 613.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 413.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 114.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 79.1T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.42e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 450.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.43e3T + 9.12e5T^{2} \) |
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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
−10.32140113845710982564089766998, −9.636686395186789825868077336809, −8.686089915407460075227237399907, −6.87759025341050580412273402887, −6.58441733249264846757167061942, −5.33496753181973848550727468372, −4.54237634283927126290037434941, −3.24236934524999340222888111339, −2.44331582793314423568838919208, 0,
2.44331582793314423568838919208, 3.24236934524999340222888111339, 4.54237634283927126290037434941, 5.33496753181973848550727468372, 6.58441733249264846757167061942, 6.87759025341050580412273402887, 8.686089915407460075227237399907, 9.636686395186789825868077336809, 10.32140113845710982564089766998
