| L(s) = 1 | + (−2 + 3.46i)2-s + (−2.43 − 15.3i)3-s + (−7.99 − 13.8i)4-s + (−20.8 − 36.0i)5-s + (58.2 + 22.3i)6-s + (101. − 176. i)7-s + 63.9·8-s + (−231. + 74.9i)9-s + 166.·10-s + (−235. + 407. i)11-s + (−193. + 156. i)12-s + (−241. − 418. i)13-s + (406. + 704. i)14-s + (−504. + 407. i)15-s + (−128 + 221. i)16-s + 1.25e3·17-s + ⋯ |
| L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.156 − 0.987i)3-s + (−0.249 − 0.433i)4-s + (−0.372 − 0.644i)5-s + (0.660 + 0.253i)6-s + (0.784 − 1.35i)7-s + 0.353·8-s + (−0.951 + 0.308i)9-s + 0.526·10-s + (−0.585 + 1.01i)11-s + (−0.388 + 0.314i)12-s + (−0.396 − 0.686i)13-s + (0.554 + 0.960i)14-s + (−0.578 + 0.468i)15-s + (−0.125 + 0.216i)16-s + 1.05·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.138 + 0.990i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.138 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.711464 - 0.618852i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.711464 - 0.618852i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (2 - 3.46i)T \) |
| 3 | \( 1 + (2.43 + 15.3i)T \) |
| good | 5 | \( 1 + (20.8 + 36.0i)T + (-1.56e3 + 2.70e3i)T^{2} \) |
| 7 | \( 1 + (-101. + 176. i)T + (-8.40e3 - 1.45e4i)T^{2} \) |
| 11 | \( 1 + (235. - 407. i)T + (-8.05e4 - 1.39e5i)T^{2} \) |
| 13 | \( 1 + (241. + 418. i)T + (-1.85e5 + 3.21e5i)T^{2} \) |
| 17 | \( 1 - 1.25e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.97e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + (239. + 414. i)T + (-3.21e6 + 5.57e6i)T^{2} \) |
| 29 | \( 1 + (-580. + 1.00e3i)T + (-1.02e7 - 1.77e7i)T^{2} \) |
| 31 | \( 1 + (-1.18e3 - 2.05e3i)T + (-1.43e7 + 2.47e7i)T^{2} \) |
| 37 | \( 1 - 8.18e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + (8.75e3 + 1.51e4i)T + (-5.79e7 + 1.00e8i)T^{2} \) |
| 43 | \( 1 + (1.14e4 - 1.98e4i)T + (-7.35e7 - 1.27e8i)T^{2} \) |
| 47 | \( 1 + (-8.68e3 + 1.50e4i)T + (-1.14e8 - 1.98e8i)T^{2} \) |
| 53 | \( 1 + 5.39e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + (-2.22e4 - 3.85e4i)T + (-3.57e8 + 6.19e8i)T^{2} \) |
| 61 | \( 1 + (2.08e3 - 3.60e3i)T + (-4.22e8 - 7.31e8i)T^{2} \) |
| 67 | \( 1 + (1.22e3 + 2.12e3i)T + (-6.75e8 + 1.16e9i)T^{2} \) |
| 71 | \( 1 - 2.18e3T + 1.80e9T^{2} \) |
| 73 | \( 1 - 3.03e3T + 2.07e9T^{2} \) |
| 79 | \( 1 + (-2.52e4 + 4.37e4i)T + (-1.53e9 - 2.66e9i)T^{2} \) |
| 83 | \( 1 + (-2.59e4 + 4.48e4i)T + (-1.96e9 - 3.41e9i)T^{2} \) |
| 89 | \( 1 + 2.01e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + (4.02e4 - 6.96e4i)T + (-4.29e9 - 7.43e9i)T^{2} \) |
| 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
−17.42049655817944042852234461490, −16.40424592337537953139432736475, −14.65174390824673245448867612688, −13.42934193235437147742669244323, −12.05497591550853884360819620928, −10.25892326133427922183191293679, −8.016668984762849669723093966124, −7.33103831796893653089430074620, −5.04791351640243052345209877504, −0.916430588039136972737422307563,
3.06793779425881685658265795884, 5.34074953897671859287786828179, 8.209909422166116065329542666778, 9.616811875477401714869018676737, 11.16920258402269954026601765740, 11.87180437707102884142481858592, 14.22993117073027092372100101866, 15.38581037304592269836720214771, 16.62160747948628093428452776286, 18.23774701279597346327758414378
