| L(s) = 1 | + (−0.626 − 1.92i)2-s + (3.14 − 2.28i)4-s + (−3.12 − 10.7i)5-s − 16.4·7-s + (−19.4 − 14.1i)8-s + (−18.7 + 12.7i)10-s + (−1.41 − 4.34i)11-s + (−17.0 + 52.4i)13-s + (10.2 + 31.6i)14-s + (−5.47 + 16.8i)16-s + (−1.72 − 1.25i)17-s + (22.2 + 16.1i)19-s + (−34.3 − 26.6i)20-s + (−7.48 + 5.43i)22-s + (−30.4 − 93.5i)23-s + ⋯ |
| L(s) = 1 | + (−0.221 − 0.681i)2-s + (0.393 − 0.285i)4-s + (−0.279 − 0.960i)5-s − 0.887·7-s + (−0.861 − 0.626i)8-s + (−0.592 + 0.402i)10-s + (−0.0386 − 0.119i)11-s + (−0.363 + 1.11i)13-s + (0.196 + 0.605i)14-s + (−0.0855 + 0.263i)16-s + (−0.0246 − 0.0179i)17-s + (0.268 + 0.194i)19-s + (−0.384 − 0.298i)20-s + (−0.0725 + 0.0527i)22-s + (−0.275 − 0.848i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.588 - 0.808i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.588 - 0.808i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.212264 + 0.416977i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.212264 + 0.416977i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (3.12 + 10.7i)T \) |
| good | 2 | \( 1 + (0.626 + 1.92i)T + (-6.47 + 4.70i)T^{2} \) |
| 7 | \( 1 + 16.4T + 343T^{2} \) |
| 11 | \( 1 + (1.41 + 4.34i)T + (-1.07e3 + 782. i)T^{2} \) |
| 13 | \( 1 + (17.0 - 52.4i)T + (-1.77e3 - 1.29e3i)T^{2} \) |
| 17 | \( 1 + (1.72 + 1.25i)T + (1.51e3 + 4.67e3i)T^{2} \) |
| 19 | \( 1 + (-22.2 - 16.1i)T + (2.11e3 + 6.52e3i)T^{2} \) |
| 23 | \( 1 + (30.4 + 93.5i)T + (-9.84e3 + 7.15e3i)T^{2} \) |
| 29 | \( 1 + (-2.70 + 1.96i)T + (7.53e3 - 2.31e4i)T^{2} \) |
| 31 | \( 1 + (-54.6 - 39.6i)T + (9.20e3 + 2.83e4i)T^{2} \) |
| 37 | \( 1 + (56.4 - 173. i)T + (-4.09e4 - 2.97e4i)T^{2} \) |
| 41 | \( 1 + (-11.7 + 36.2i)T + (-5.57e4 - 4.05e4i)T^{2} \) |
| 43 | \( 1 + 252.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (35.6 - 25.9i)T + (3.20e4 - 9.87e4i)T^{2} \) |
| 53 | \( 1 + (163. - 119. i)T + (4.60e4 - 1.41e5i)T^{2} \) |
| 59 | \( 1 + (-36.4 + 112. i)T + (-1.66e5 - 1.20e5i)T^{2} \) |
| 61 | \( 1 + (248. + 763. i)T + (-1.83e5 + 1.33e5i)T^{2} \) |
| 67 | \( 1 + (-523. - 380. i)T + (9.29e4 + 2.86e5i)T^{2} \) |
| 71 | \( 1 + (798. - 580. i)T + (1.10e5 - 3.40e5i)T^{2} \) |
| 73 | \( 1 + (256. + 790. i)T + (-3.14e5 + 2.28e5i)T^{2} \) |
| 79 | \( 1 + (-822. + 597. i)T + (1.52e5 - 4.68e5i)T^{2} \) |
| 83 | \( 1 + (926. + 673. i)T + (1.76e5 + 5.43e5i)T^{2} \) |
| 89 | \( 1 + (349. + 1.07e3i)T + (-5.70e5 + 4.14e5i)T^{2} \) |
| 97 | \( 1 + (-457. + 332. i)T + (2.82e5 - 8.68e5i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.30936415307538653127285599582, −10.07238931892817189216088724605, −9.447099137775718807107072626525, −8.477140189742609049068828106324, −6.99421933834055183980472928044, −6.04625608421563602972376612469, −4.60032564846168128633599581008, −3.21840611796127321146577800157, −1.70810121758025318503501713216, −0.19311886422721754163099770193,
2.64256545950559556718579446759, 3.53507112433546230218710094075, 5.56614060537540131497194258000, 6.54382603469615926759774819739, 7.35619213332734184462341968984, 8.134533889478604554484665022719, 9.511141240501878605009645000930, 10.44387469527345556493412024698, 11.46612452913854594171358638636, 12.34209783399219645567205995763
