| 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
−12.34209783399219645567205995763, −11.46612452913854594171358638636, −10.44387469527345556493412024698, −9.511141240501878605009645000930, −8.134533889478604554484665022719, −7.35619213332734184462341968984, −6.54382603469615926759774819739, −5.56614060537540131497194258000, −3.53507112433546230218710094075, −2.64256545950559556718579446759,
0.19311886422721754163099770193, 1.70810121758025318503501713216, 3.21840611796127321146577800157, 4.60032564846168128633599581008, 6.04625608421563602972376612469, 6.99421933834055183980472928044, 8.477140189742609049068828106324, 9.447099137775718807107072626525, 10.07238931892817189216088724605, 11.30936415307538653127285599582
