| L(s) = 1 | + (−4.95 + 1.60i)2-s + (−1.70 − 2.34i)3-s + (15.4 − 11.2i)4-s + (11.0 + 1.96i)5-s + (12.2 + 8.87i)6-s − 24.5i·7-s + (−34.1 + 46.9i)8-s + (5.74 − 17.6i)9-s + (−57.6 + 7.96i)10-s + (−0.0967 − 0.297i)11-s + (−52.7 − 17.1i)12-s + (39.6 + 12.8i)13-s + (39.5 + 121. i)14-s + (−14.1 − 29.1i)15-s + (46.1 − 141. i)16-s + (3.64 − 5.01i)17-s + ⋯ |
| L(s) = 1 | + (−1.75 + 0.569i)2-s + (−0.327 − 0.451i)3-s + (1.93 − 1.40i)4-s + (0.984 + 0.176i)5-s + (0.830 + 0.603i)6-s − 1.32i·7-s + (−1.50 + 2.07i)8-s + (0.212 − 0.655i)9-s + (−1.82 + 0.251i)10-s + (−0.00265 − 0.00816i)11-s + (−1.26 − 0.412i)12-s + (0.846 + 0.275i)13-s + (0.755 + 2.32i)14-s + (−0.243 − 0.501i)15-s + (0.720 − 2.21i)16-s + (0.0520 − 0.0715i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.822 + 0.569i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.822 + 0.569i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.539007 - 0.168310i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.539007 - 0.168310i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-11.0 - 1.96i)T \) |
| good | 2 | \( 1 + (4.95 - 1.60i)T + (6.47 - 4.70i)T^{2} \) |
| 3 | \( 1 + (1.70 + 2.34i)T + (-8.34 + 25.6i)T^{2} \) |
| 7 | \( 1 + 24.5iT - 343T^{2} \) |
| 11 | \( 1 + (0.0967 + 0.297i)T + (-1.07e3 + 782. i)T^{2} \) |
| 13 | \( 1 + (-39.6 - 12.8i)T + (1.77e3 + 1.29e3i)T^{2} \) |
| 17 | \( 1 + (-3.64 + 5.01i)T + (-1.51e3 - 4.67e3i)T^{2} \) |
| 19 | \( 1 + (69.4 + 50.4i)T + (2.11e3 + 6.52e3i)T^{2} \) |
| 23 | \( 1 + (-40.3 + 13.1i)T + (9.84e3 - 7.15e3i)T^{2} \) |
| 29 | \( 1 + (186. - 135. i)T + (7.53e3 - 2.31e4i)T^{2} \) |
| 31 | \( 1 + (-195. - 141. i)T + (9.20e3 + 2.83e4i)T^{2} \) |
| 37 | \( 1 + (-137. - 44.7i)T + (4.09e4 + 2.97e4i)T^{2} \) |
| 41 | \( 1 + (-7.44 + 22.9i)T + (-5.57e4 - 4.05e4i)T^{2} \) |
| 43 | \( 1 - 259. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + (179. + 246. i)T + (-3.20e4 + 9.87e4i)T^{2} \) |
| 53 | \( 1 + (-221. - 305. i)T + (-4.60e4 + 1.41e5i)T^{2} \) |
| 59 | \( 1 + (-69.6 + 214. i)T + (-1.66e5 - 1.20e5i)T^{2} \) |
| 61 | \( 1 + (-166. - 512. i)T + (-1.83e5 + 1.33e5i)T^{2} \) |
| 67 | \( 1 + (34.5 - 47.5i)T + (-9.29e4 - 2.86e5i)T^{2} \) |
| 71 | \( 1 + (29.2 - 21.2i)T + (1.10e5 - 3.40e5i)T^{2} \) |
| 73 | \( 1 + (779. - 253. i)T + (3.14e5 - 2.28e5i)T^{2} \) |
| 79 | \( 1 + (-249. + 181. i)T + (1.52e5 - 4.68e5i)T^{2} \) |
| 83 | \( 1 + (-249. + 342. i)T + (-1.76e5 - 5.43e5i)T^{2} \) |
| 89 | \( 1 + (-139. - 428. i)T + (-5.70e5 + 4.14e5i)T^{2} \) |
| 97 | \( 1 + (-232. - 320. i)T + (-2.82e5 + 8.68e5i)T^{2} \) |
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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
−17.26059351354462371015590747495, −16.39688952276107530211326993007, −14.82403016950594955188361769562, −13.29200963967238981080186634715, −11.11296483755803646473232560465, −10.14238528953088191516709235648, −8.905056751739496004539371820263, −7.14741070975723145225254606284, −6.32450595883918361127253186565, −1.16816836229897334952985639712,
2.13846219593790320002541433132, 5.95747910473422333104742238574, 8.216199119604827517083210849172, 9.341445986747722996946984400136, 10.36244983420619419382171223601, 11.50242438265350598236654501636, 12.98855083845474208455722644024, 15.35658108386572366255922786534, 16.48484833383705377240493638813, 17.36804233218329229588649146221
