| L(s) = 1 | + 2.03·2-s − 3.87·4-s − 5·5-s + 27.2·7-s − 24.1·8-s − 10.1·10-s − 11·11-s + 14.8·13-s + 55.2·14-s − 17.9·16-s − 91.7·17-s − 12.7·19-s + 19.3·20-s − 22.3·22-s + 10.2·23-s + 25·25-s + 30.1·26-s − 105.·28-s − 153.·29-s − 115.·31-s + 156.·32-s − 186.·34-s − 136.·35-s + 201.·37-s − 25.9·38-s + 120.·40-s − 398.·41-s + ⋯ |
| L(s) = 1 | + 0.718·2-s − 0.484·4-s − 0.447·5-s + 1.46·7-s − 1.06·8-s − 0.321·10-s − 0.301·11-s + 0.316·13-s + 1.05·14-s − 0.280·16-s − 1.30·17-s − 0.154·19-s + 0.216·20-s − 0.216·22-s + 0.0931·23-s + 0.200·25-s + 0.227·26-s − 0.711·28-s − 0.985·29-s − 0.669·31-s + 0.864·32-s − 0.940·34-s − 0.657·35-s + 0.896·37-s − 0.110·38-s + 0.476·40-s − 1.51·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 5T \) |
| 11 | \( 1 + 11T \) |
| good | 2 | \( 1 - 2.03T + 8T^{2} \) |
| 7 | \( 1 - 27.2T + 343T^{2} \) |
| 13 | \( 1 - 14.8T + 2.19e3T^{2} \) |
| 17 | \( 1 + 91.7T + 4.91e3T^{2} \) |
| 19 | \( 1 + 12.7T + 6.85e3T^{2} \) |
| 23 | \( 1 - 10.2T + 1.21e4T^{2} \) |
| 29 | \( 1 + 153.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 115.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 201.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 398.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 438.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 372.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 454.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 766.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 357.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 947.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 568.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 503.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 386.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 754.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 848.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 823.T + 9.12e5T^{2} \) |
| 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
−10.22068742103137571717697788823, −8.912393114872261382094207256758, −8.402055470608284120295241652709, −7.40632779396253642452499622418, −6.13498339692469618027679983289, −4.98778275657237485142534478703, −4.49938805407025416110289025137, −3.37086045928477637959401006571, −1.80822769475557872480782750380, 0,
1.80822769475557872480782750380, 3.37086045928477637959401006571, 4.49938805407025416110289025137, 4.98778275657237485142534478703, 6.13498339692469618027679983289, 7.40632779396253642452499622418, 8.402055470608284120295241652709, 8.912393114872261382094207256758, 10.22068742103137571717697788823
