| L(s) = 1 | + (−1.94 + 0.462i)2-s + (2.33 − 1.87i)3-s + (3.57 − 1.79i)4-s + (−2.39 − 4.38i)5-s + (−3.68 + 4.73i)6-s + (4.78 − 4.78i)7-s + (−6.12 + 5.15i)8-s + (1.93 − 8.78i)9-s + (6.68 + 7.43i)10-s + (−0.198 − 0.143i)11-s + (4.97 − 10.9i)12-s + (0.392 − 2.47i)13-s + (−7.10 + 11.5i)14-s + (−13.8 − 5.76i)15-s + (9.52 − 12.8i)16-s + (−3.53 − 6.93i)17-s + ⋯ |
| L(s) = 1 | + (−0.972 + 0.231i)2-s + (0.779 − 0.626i)3-s + (0.893 − 0.449i)4-s + (−0.478 − 0.877i)5-s + (−0.613 + 0.789i)6-s + (0.684 − 0.684i)7-s + (−0.765 + 0.643i)8-s + (0.215 − 0.976i)9-s + (0.668 + 0.743i)10-s + (−0.0180 − 0.0130i)11-s + (0.414 − 0.910i)12-s + (0.0301 − 0.190i)13-s + (−0.507 + 0.823i)14-s + (−0.923 − 0.384i)15-s + (0.595 − 0.803i)16-s + (−0.207 − 0.407i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.476 + 0.879i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.476 + 0.879i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.603433 - 1.01324i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.603433 - 1.01324i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.94 - 0.462i)T \) |
| 3 | \( 1 + (-2.33 + 1.87i)T \) |
| 5 | \( 1 + (2.39 + 4.38i)T \) |
| good | 7 | \( 1 + (-4.78 + 4.78i)T - 49iT^{2} \) |
| 11 | \( 1 + (0.198 + 0.143i)T + (37.3 + 115. i)T^{2} \) |
| 13 | \( 1 + (-0.392 + 2.47i)T + (-160. - 52.2i)T^{2} \) |
| 17 | \( 1 + (3.53 + 6.93i)T + (-169. + 233. i)T^{2} \) |
| 19 | \( 1 + (2.59 - 7.98i)T + (-292. - 212. i)T^{2} \) |
| 23 | \( 1 + (4.23 - 0.670i)T + (503. - 163. i)T^{2} \) |
| 29 | \( 1 + (-8.46 - 26.0i)T + (-680. + 494. i)T^{2} \) |
| 31 | \( 1 + (30.0 + 9.76i)T + (777. + 564. i)T^{2} \) |
| 37 | \( 1 + (-42.6 - 6.76i)T + (1.30e3 + 423. i)T^{2} \) |
| 41 | \( 1 + (3.16 + 4.36i)T + (-519. + 1.59e3i)T^{2} \) |
| 43 | \( 1 + (54.3 + 54.3i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (-31.7 + 62.3i)T + (-1.29e3 - 1.78e3i)T^{2} \) |
| 53 | \( 1 + (-32.1 + 63.0i)T + (-1.65e3 - 2.27e3i)T^{2} \) |
| 59 | \( 1 + (53.1 + 73.1i)T + (-1.07e3 + 3.31e3i)T^{2} \) |
| 61 | \( 1 + (2.05 + 1.49i)T + (1.14e3 + 3.53e3i)T^{2} \) |
| 67 | \( 1 + (-39.7 - 77.9i)T + (-2.63e3 + 3.63e3i)T^{2} \) |
| 71 | \( 1 + (-6.51 - 20.0i)T + (-4.07e3 + 2.96e3i)T^{2} \) |
| 73 | \( 1 + (-94.4 + 14.9i)T + (5.06e3 - 1.64e3i)T^{2} \) |
| 79 | \( 1 + (-39.6 - 121. i)T + (-5.04e3 + 3.66e3i)T^{2} \) |
| 83 | \( 1 + (34.9 + 68.5i)T + (-4.04e3 + 5.57e3i)T^{2} \) |
| 89 | \( 1 + (-43.8 - 31.8i)T + (2.44e3 + 7.53e3i)T^{2} \) |
| 97 | \( 1 + (-7.39 + 14.5i)T + (-5.53e3 - 7.61e3i)T^{2} \) |
| 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
−11.20296334127901171849805114519, −10.03710217052564940944828001944, −9.024347064437190501463785332374, −8.292344086420377887974591276424, −7.63954542988369232778145071678, −6.78265540486529826658794726829, −5.25674847832492081567261574827, −3.70859081224349820845815381366, −1.93470122084860506580888160468, −0.71647268622539849077187042938,
2.08188792972789636244803657458, 3.08086127236620868489501877813, 4.37486408346487597678409342300, 6.13960162445127081856101733936, 7.45492599510108620581138916698, 8.139645011534455931569847119167, 8.990275299981050682837372170754, 9.885113203182355652351700786546, 10.87602295164966128560949446893, 11.36535663121625567039781887775
