| L(s) = 1 | + (−4.17 + 1.35i)2-s + (−0.0707 − 0.0973i)3-s + (−10.3 + 7.48i)4-s + (26.6 − 49.1i)5-s + (0.427 + 0.310i)6-s − 51.4i·7-s + (115. − 158. i)8-s + (75.0 − 231. i)9-s + (−44.7 + 241. i)10-s + (−118. − 365. i)11-s + (1.45 + 0.473i)12-s + (306. + 99.4i)13-s + (69.7 + 214. i)14-s + (−6.66 + 0.877i)15-s + (−140. + 432. i)16-s + (−1.11e3 + 1.54e3i)17-s + ⋯ |
| L(s) = 1 | + (−0.737 + 0.239i)2-s + (−0.00453 − 0.00624i)3-s + (−0.321 + 0.233i)4-s + (0.477 − 0.878i)5-s + (0.00484 + 0.00352i)6-s − 0.396i·7-s + (0.637 − 0.877i)8-s + (0.308 − 0.950i)9-s + (−0.141 + 0.762i)10-s + (−0.295 − 0.910i)11-s + (0.00292 + 0.000949i)12-s + (0.502 + 0.163i)13-s + (0.0951 + 0.292i)14-s + (−0.00765 + 0.00100i)15-s + (−0.137 + 0.421i)16-s + (−0.939 + 1.29i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.318 + 0.947i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.318 + 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.680995 - 0.489469i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.680995 - 0.489469i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-26.6 + 49.1i)T \) |
| good | 2 | \( 1 + (4.17 - 1.35i)T + (25.8 - 18.8i)T^{2} \) |
| 3 | \( 1 + (0.0707 + 0.0973i)T + (-75.0 + 231. i)T^{2} \) |
| 7 | \( 1 + 51.4iT - 1.68e4T^{2} \) |
| 11 | \( 1 + (118. + 365. i)T + (-1.30e5 + 9.46e4i)T^{2} \) |
| 13 | \( 1 + (-306. - 99.4i)T + (3.00e5 + 2.18e5i)T^{2} \) |
| 17 | \( 1 + (1.11e3 - 1.54e3i)T + (-4.38e5 - 1.35e6i)T^{2} \) |
| 19 | \( 1 + (1.53e3 + 1.11e3i)T + (7.65e5 + 2.35e6i)T^{2} \) |
| 23 | \( 1 + (-3.77e3 + 1.22e3i)T + (5.20e6 - 3.78e6i)T^{2} \) |
| 29 | \( 1 + (-3.48e3 + 2.53e3i)T + (6.33e6 - 1.95e7i)T^{2} \) |
| 31 | \( 1 + (4.47e3 + 3.24e3i)T + (8.84e6 + 2.72e7i)T^{2} \) |
| 37 | \( 1 + (-1.80e3 - 586. i)T + (5.61e7 + 4.07e7i)T^{2} \) |
| 41 | \( 1 + (3.67e3 - 1.13e4i)T + (-9.37e7 - 6.80e7i)T^{2} \) |
| 43 | \( 1 + 3.12e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 + (-6.37e3 - 8.78e3i)T + (-7.08e7 + 2.18e8i)T^{2} \) |
| 53 | \( 1 + (-5.71e3 - 7.85e3i)T + (-1.29e8 + 3.97e8i)T^{2} \) |
| 59 | \( 1 + (2.86e3 - 8.80e3i)T + (-5.78e8 - 4.20e8i)T^{2} \) |
| 61 | \( 1 + (-2.54e3 - 7.82e3i)T + (-6.83e8 + 4.96e8i)T^{2} \) |
| 67 | \( 1 + (4.76e3 - 6.55e3i)T + (-4.17e8 - 1.28e9i)T^{2} \) |
| 71 | \( 1 + (-5.55e4 + 4.03e4i)T + (5.57e8 - 1.71e9i)T^{2} \) |
| 73 | \( 1 + (-4.39e4 + 1.42e4i)T + (1.67e9 - 1.21e9i)T^{2} \) |
| 79 | \( 1 + (4.41e4 - 3.21e4i)T + (9.50e8 - 2.92e9i)T^{2} \) |
| 83 | \( 1 + (1.29e4 - 1.77e4i)T + (-1.21e9 - 3.74e9i)T^{2} \) |
| 89 | \( 1 + (-3.56e4 - 1.09e5i)T + (-4.51e9 + 3.28e9i)T^{2} \) |
| 97 | \( 1 + (-1.15e4 - 1.59e4i)T + (-2.65e9 + 8.16e9i)T^{2} \) |
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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
−16.72205656617849589879073722326, −15.35924169059666380592819232822, −13.44384655765251936218584809274, −12.77572104449356204554383850345, −10.72850532717470872174390376379, −9.183708465702119806059420780463, −8.428344266088973092667128830104, −6.47025669855160944116505809993, −4.22174255361794549210033346424, −0.76869683055537766016730285760,
2.13101893961252215546067961050, 5.10909867949642820119962546585, 7.19091173753142303700106769248, 8.895478812943560246521209904664, 10.21372602581189725181506289155, 11.05461297138340951535411798287, 13.12861374931504140490055579182, 14.25710585979778043754393575190, 15.54841343552968531155845695443, 17.17568751745403511208310390013
