| L(s) = 1 | + (−3.23 + 2.35i)2-s + (2.78 − 8.55i)3-s + (4.94 − 15.2i)4-s + (53.0 + 17.5i)5-s + (11.1 + 34.2i)6-s − 17.5·7-s + (19.7 + 60.8i)8-s + (−65.5 − 47.6i)9-s + (−213. + 67.9i)10-s + (−601. + 436. i)11-s + (−116. − 84.6i)12-s + (365. + 265. i)13-s + (56.9 − 41.3i)14-s + (297. − 405. i)15-s + (−207. − 150. i)16-s + (−122. − 377. i)17-s + ⋯ |
| L(s) = 1 | + (−0.572 + 0.415i)2-s + (0.178 − 0.549i)3-s + (0.154 − 0.475i)4-s + (0.949 + 0.314i)5-s + (0.126 + 0.388i)6-s − 0.135·7-s + (0.109 + 0.336i)8-s + (−0.269 − 0.195i)9-s + (−0.673 + 0.214i)10-s + (−1.49 + 1.08i)11-s + (−0.233 − 0.169i)12-s + (0.600 + 0.436i)13-s + (0.0776 − 0.0564i)14-s + (0.341 − 0.465i)15-s + (−0.202 − 0.146i)16-s + (−0.102 − 0.316i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.415 - 0.909i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.415 - 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.063892536\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.063892536\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (3.23 - 2.35i)T \) |
| 3 | \( 1 + (-2.78 + 8.55i)T \) |
| 5 | \( 1 + (-53.0 - 17.5i)T \) |
| good | 7 | \( 1 + 17.5T + 1.68e4T^{2} \) |
| 11 | \( 1 + (601. - 436. i)T + (4.97e4 - 1.53e5i)T^{2} \) |
| 13 | \( 1 + (-365. - 265. i)T + (1.14e5 + 3.53e5i)T^{2} \) |
| 17 | \( 1 + (122. + 377. i)T + (-1.14e6 + 8.34e5i)T^{2} \) |
| 19 | \( 1 + (63.3 + 194. i)T + (-2.00e6 + 1.45e6i)T^{2} \) |
| 23 | \( 1 + (1.24e3 - 903. i)T + (1.98e6 - 6.12e6i)T^{2} \) |
| 29 | \( 1 + (572. - 1.76e3i)T + (-1.65e7 - 1.20e7i)T^{2} \) |
| 31 | \( 1 + (-725. - 2.23e3i)T + (-2.31e7 + 1.68e7i)T^{2} \) |
| 37 | \( 1 + (-3.62e3 - 2.63e3i)T + (2.14e7 + 6.59e7i)T^{2} \) |
| 41 | \( 1 + (-3.85e3 - 2.80e3i)T + (3.58e7 + 1.10e8i)T^{2} \) |
| 43 | \( 1 + 1.70e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + (8.01e3 - 2.46e4i)T + (-1.85e8 - 1.34e8i)T^{2} \) |
| 53 | \( 1 + (8.38e3 - 2.57e4i)T + (-3.38e8 - 2.45e8i)T^{2} \) |
| 59 | \( 1 + (-2.75e4 - 2.00e4i)T + (2.20e8 + 6.79e8i)T^{2} \) |
| 61 | \( 1 + (-1.69e4 + 1.23e4i)T + (2.60e8 - 8.03e8i)T^{2} \) |
| 67 | \( 1 + (6.39e3 + 1.96e4i)T + (-1.09e9 + 7.93e8i)T^{2} \) |
| 71 | \( 1 + (-2.89e3 + 8.92e3i)T + (-1.45e9 - 1.06e9i)T^{2} \) |
| 73 | \( 1 + (8.41e3 - 6.11e3i)T + (6.40e8 - 1.97e9i)T^{2} \) |
| 79 | \( 1 + (2.05e4 - 6.33e4i)T + (-2.48e9 - 1.80e9i)T^{2} \) |
| 83 | \( 1 + (-3.58e4 - 1.10e5i)T + (-3.18e9 + 2.31e9i)T^{2} \) |
| 89 | \( 1 + (-4.42e4 + 3.21e4i)T + (1.72e9 - 5.31e9i)T^{2} \) |
| 97 | \( 1 + (-4.02e4 + 1.23e5i)T + (-6.94e9 - 5.04e9i)T^{2} \) |
| 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
−12.67334813192084648951016330939, −11.20050668675271019748188056574, −10.15743364035776655247797866842, −9.394675315987159434876204426934, −8.143353134884929198544743447757, −7.14884920993775071027484072306, −6.20011644351137843360137101831, −5.00576344029844102848825916878, −2.70738468305551738206517864898, −1.55759032703198359667675710523,
0.40427810582746516541916042896, 2.19444706691325117436214895193, 3.42699788560316521102059976670, 5.16725913247560603799132164333, 6.20249422829456341335321420117, 8.061561400658067068394758629413, 8.690238294939028379696270381536, 9.954049546589324423280719012526, 10.46231229909227129710635446462, 11.48226568237087652854354105157
