| L(s) = 1 | + (0.874 + 0.0500i)2-s + (1.74 − 5.35i)3-s + (−7.18 − 0.824i)4-s + (9.63 + 18.2i)5-s + (1.79 − 4.59i)6-s + (26.7 + 11.3i)7-s + (−13.1 − 2.27i)8-s + (−3.81 − 2.76i)9-s + (7.51 + 16.4i)10-s + (−36.1 − 5.23i)11-s + (−16.9 + 37.0i)12-s + (45.9 − 25.9i)13-s + (22.8 + 11.2i)14-s + (114. − 19.8i)15-s + (44.9 + 10.4i)16-s + (65.0 − 23.1i)17-s + ⋯ |
| L(s) = 1 | + (0.309 + 0.0176i)2-s + (0.334 − 1.03i)3-s + (−0.898 − 0.103i)4-s + (0.861 + 1.63i)5-s + (0.121 − 0.312i)6-s + (1.44 + 0.611i)7-s + (−0.581 − 0.100i)8-s + (−0.141 − 0.102i)9-s + (0.237 + 0.520i)10-s + (−0.989 − 0.143i)11-s + (−0.406 + 0.891i)12-s + (0.979 − 0.553i)13-s + (0.436 + 0.214i)14-s + (1.97 − 0.341i)15-s + (0.702 + 0.163i)16-s + (0.927 − 0.330i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 - 0.147i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.989 - 0.147i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.18839 + 0.162572i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.18839 + 0.162572i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + (36.1 + 5.23i)T \) |
| good | 2 | \( 1 + (-0.874 - 0.0500i)T + (7.94 + 0.911i)T^{2} \) |
| 3 | \( 1 + (-1.74 + 5.35i)T + (-21.8 - 15.8i)T^{2} \) |
| 5 | \( 1 + (-9.63 - 18.2i)T + (-70.5 + 103. i)T^{2} \) |
| 7 | \( 1 + (-26.7 - 11.3i)T + (239. + 245. i)T^{2} \) |
| 13 | \( 1 + (-45.9 + 25.9i)T + (1.13e3 - 1.88e3i)T^{2} \) |
| 17 | \( 1 + (-65.0 + 23.1i)T + (3.80e3 - 3.10e3i)T^{2} \) |
| 19 | \( 1 + (1.62 - 57.0i)T + (-6.84e3 - 391. i)T^{2} \) |
| 23 | \( 1 + (42.1 + 48.6i)T + (-1.73e3 + 1.20e4i)T^{2} \) |
| 29 | \( 1 + (3.60 + 5.26i)T + (-8.84e3 + 2.27e4i)T^{2} \) |
| 31 | \( 1 + (-10.3 + 51.2i)T + (-2.74e4 - 1.15e4i)T^{2} \) |
| 37 | \( 1 + (234. - 215. i)T + (4.33e3 - 5.04e4i)T^{2} \) |
| 41 | \( 1 + (52.4 + 199. i)T + (-6.00e4 + 3.38e4i)T^{2} \) |
| 43 | \( 1 + (7.14 + 49.6i)T + (-7.62e4 + 2.23e4i)T^{2} \) |
| 47 | \( 1 + (284. - 232. i)T + (2.06e4 - 1.01e5i)T^{2} \) |
| 53 | \( 1 + (-59.6 + 13.8i)T + (1.33e5 - 6.56e4i)T^{2} \) |
| 59 | \( 1 + (-168. + 641. i)T + (-1.78e5 - 1.00e5i)T^{2} \) |
| 61 | \( 1 + (780. - 44.6i)T + (2.25e5 - 2.58e4i)T^{2} \) |
| 67 | \( 1 + (-136. + 87.6i)T + (1.24e5 - 2.73e5i)T^{2} \) |
| 71 | \( 1 + (-647. + 839. i)T + (-9.09e4 - 3.46e5i)T^{2} \) |
| 73 | \( 1 + (456. - 756. i)T + (-1.81e5 - 3.44e5i)T^{2} \) |
| 79 | \( 1 + (-302. + 311. i)T + (-1.40e4 - 4.92e5i)T^{2} \) |
| 83 | \( 1 + (-50.5 + 588. i)T + (-5.63e5 - 9.75e4i)T^{2} \) |
| 89 | \( 1 + (58.3 + 17.1i)T + (5.93e5 + 3.81e5i)T^{2} \) |
| 97 | \( 1 + (-362. + 686. i)T + (-5.15e5 - 7.53e5i)T^{2} \) |
| 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
−13.35217056706433344795956432631, −12.20374253354512778403999966240, −10.86252816788176273077349153333, −10.02889097427399697163508328369, −8.367602836285931792438495352866, −7.70471802396981957628420615092, −6.18792021509038313701876308152, −5.26026189774583845051743570914, −3.11343934117316830656719964297, −1.76012757327328339188445182708,
1.30040771417087879266430032572, 3.96864045567250193147567518945, 4.83059056475820477051856578648, 5.43768379757437132526370155057, 8.083250044478326474839535780684, 8.775017785368032570618781263685, 9.644836003606483101631991253597, 10.61601074310449814817189246988, 12.15070547844624011162567167219, 13.26357360239396530929931823359
