| L(s) = 1 | + (0.101 − 0.311i)2-s + (−1.02 + 4.83i)3-s + (3.14 + 2.28i)4-s + (−3.67 − 6.37i)5-s + (1.40 + 0.808i)6-s + (6.74 − 3.00i)7-s + (2.08 − 1.51i)8-s + (−14.1 − 6.28i)9-s + (−2.35 + 0.500i)10-s + (−3.04 − 0.320i)11-s + (−14.3 + 12.8i)12-s + (−1.63 − 1.47i)13-s + (−0.252 − 2.40i)14-s + (34.6 − 11.2i)15-s + (4.55 + 14.0i)16-s + (1.91 − 0.200i)17-s + ⋯ |
| L(s) = 1 | + (0.0505 − 0.155i)2-s + (−0.342 + 1.61i)3-s + (0.787 + 0.572i)4-s + (−0.735 − 1.27i)5-s + (0.233 + 0.134i)6-s + (0.964 − 0.429i)7-s + (0.261 − 0.189i)8-s + (−1.56 − 0.697i)9-s + (−0.235 + 0.0500i)10-s + (−0.276 − 0.0290i)11-s + (−1.19 + 1.07i)12-s + (−0.125 − 0.113i)13-s + (−0.0180 − 0.171i)14-s + (2.30 − 0.749i)15-s + (0.284 + 0.875i)16-s + (0.112 − 0.0118i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.712 - 0.702i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.712 - 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.930004 + 0.381316i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.930004 + 0.381316i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 31 | \( 1 + (25.2 - 17.9i)T \) |
| good | 2 | \( 1 + (-0.101 + 0.311i)T + (-3.23 - 2.35i)T^{2} \) |
| 3 | \( 1 + (1.02 - 4.83i)T + (-8.22 - 3.66i)T^{2} \) |
| 5 | \( 1 + (3.67 + 6.37i)T + (-12.5 + 21.6i)T^{2} \) |
| 7 | \( 1 + (-6.74 + 3.00i)T + (32.7 - 36.4i)T^{2} \) |
| 11 | \( 1 + (3.04 + 0.320i)T + (118. + 25.1i)T^{2} \) |
| 13 | \( 1 + (1.63 + 1.47i)T + (17.6 + 168. i)T^{2} \) |
| 17 | \( 1 + (-1.91 + 0.200i)T + (282. - 60.0i)T^{2} \) |
| 19 | \( 1 + (17.7 + 19.7i)T + (-37.7 + 359. i)T^{2} \) |
| 23 | \( 1 + (-16.3 - 22.5i)T + (-163. + 503. i)T^{2} \) |
| 29 | \( 1 + (44.3 + 14.4i)T + (680. + 494. i)T^{2} \) |
| 37 | \( 1 + (-38.1 - 22.0i)T + (684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-17.8 + 3.78i)T + (1.53e3 - 683. i)T^{2} \) |
| 43 | \( 1 + (-19.8 + 17.9i)T + (193. - 1.83e3i)T^{2} \) |
| 47 | \( 1 + (1.60 + 4.94i)T + (-1.78e3 + 1.29e3i)T^{2} \) |
| 53 | \( 1 + (9.98 - 22.4i)T + (-1.87e3 - 2.08e3i)T^{2} \) |
| 59 | \( 1 + (-30.3 - 6.45i)T + (3.18e3 + 1.41e3i)T^{2} \) |
| 61 | \( 1 + 58.0iT - 3.72e3T^{2} \) |
| 67 | \( 1 + (23.1 + 40.0i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 + (-110. - 49.0i)T + (3.37e3 + 3.74e3i)T^{2} \) |
| 73 | \( 1 + (47.6 + 5.00i)T + (5.21e3 + 1.10e3i)T^{2} \) |
| 79 | \( 1 + (-80.5 + 8.46i)T + (6.10e3 - 1.29e3i)T^{2} \) |
| 83 | \( 1 + (-1.14 - 5.37i)T + (-6.29e3 + 2.80e3i)T^{2} \) |
| 89 | \( 1 + (32.2 - 44.3i)T + (-2.44e3 - 7.53e3i)T^{2} \) |
| 97 | \( 1 + (-22.3 - 16.2i)T + (2.90e3 + 8.94e3i)T^{2} \) |
| 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
−16.75816309807791314790561465926, −15.74891262634741210389818785168, −15.03394535883357814165183802361, −12.90200376527969992457225952344, −11.47602490546208493576318487133, −10.88962518687208293003220867259, −9.119049093142089569284623496781, −7.79877786265420538096102681946, −5.07355971489820982432874714911, −3.97064846678361702542942209229,
2.18560750372770419838773139536, 5.91272950107353077900365644429, 7.10075703093392900031417095849, 7.914649197238374379065633369717, 10.87157323141042445596112055392, 11.45008075715606664121973889539, 12.67219447661335678973911677695, 14.53483105883582789122764975709, 14.86165038524491882614915485504, 16.65704144172843882533419267107
