| L(s) = 1 | + (−1.23 + 3.80i)2-s + (22.7 + 16.5i)3-s + (−12.9 − 9.40i)4-s + (−50.3 + 24.2i)5-s + (−91.0 + 66.1i)6-s − 183.·7-s + (51.7 − 37.6i)8-s + (169. + 521. i)9-s + (−30.0 − 221. i)10-s + (137. − 423. i)11-s + (−139. − 428. i)12-s + (284. + 875. i)13-s + (226. − 696. i)14-s + (−1.54e3 − 280. i)15-s + (79.1 + 243. i)16-s + (−290. + 210. i)17-s + ⋯ |
| L(s) = 1 | + (−0.218 + 0.672i)2-s + (1.46 + 1.06i)3-s + (−0.404 − 0.293i)4-s + (−0.900 + 0.434i)5-s + (−1.03 + 0.750i)6-s − 1.41·7-s + (0.286 − 0.207i)8-s + (0.697 + 2.14i)9-s + (−0.0950 − 0.700i)10-s + (0.343 − 1.05i)11-s + (−0.278 − 0.858i)12-s + (0.466 + 1.43i)13-s + (0.308 − 0.949i)14-s + (−1.77 − 0.321i)15-s + (0.0772 + 0.237i)16-s + (−0.243 + 0.176i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 - 0.107i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.994 - 0.107i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.0791451 + 1.47376i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0791451 + 1.47376i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.23 - 3.80i)T \) |
| 5 | \( 1 + (50.3 - 24.2i)T \) |
| good | 3 | \( 1 + (-22.7 - 16.5i)T + (75.0 + 231. i)T^{2} \) |
| 7 | \( 1 + 183.T + 1.68e4T^{2} \) |
| 11 | \( 1 + (-137. + 423. i)T + (-1.30e5 - 9.46e4i)T^{2} \) |
| 13 | \( 1 + (-284. - 875. i)T + (-3.00e5 + 2.18e5i)T^{2} \) |
| 17 | \( 1 + (290. - 210. i)T + (4.38e5 - 1.35e6i)T^{2} \) |
| 19 | \( 1 + (-443. + 322. i)T + (7.65e5 - 2.35e6i)T^{2} \) |
| 23 | \( 1 + (1.32e3 - 4.07e3i)T + (-5.20e6 - 3.78e6i)T^{2} \) |
| 29 | \( 1 + (-842. - 612. i)T + (6.33e6 + 1.95e7i)T^{2} \) |
| 31 | \( 1 + (-5.14e3 + 3.73e3i)T + (8.84e6 - 2.72e7i)T^{2} \) |
| 37 | \( 1 + (-930. - 2.86e3i)T + (-5.61e7 + 4.07e7i)T^{2} \) |
| 41 | \( 1 + (-4.67e3 - 1.43e4i)T + (-9.37e7 + 6.80e7i)T^{2} \) |
| 43 | \( 1 + 4.23e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + (-5.55e3 - 4.03e3i)T + (7.08e7 + 2.18e8i)T^{2} \) |
| 53 | \( 1 + (-238. - 173. i)T + (1.29e8 + 3.97e8i)T^{2} \) |
| 59 | \( 1 + (1.10e3 + 3.38e3i)T + (-5.78e8 + 4.20e8i)T^{2} \) |
| 61 | \( 1 + (-5.11e3 + 1.57e4i)T + (-6.83e8 - 4.96e8i)T^{2} \) |
| 67 | \( 1 + (-3.02e4 + 2.19e4i)T + (4.17e8 - 1.28e9i)T^{2} \) |
| 71 | \( 1 + (-1.60e4 - 1.16e4i)T + (5.57e8 + 1.71e9i)T^{2} \) |
| 73 | \( 1 + (-3.98e3 + 1.22e4i)T + (-1.67e9 - 1.21e9i)T^{2} \) |
| 79 | \( 1 + (-1.43e4 - 1.04e4i)T + (9.50e8 + 2.92e9i)T^{2} \) |
| 83 | \( 1 + (-6.28e3 + 4.56e3i)T + (1.21e9 - 3.74e9i)T^{2} \) |
| 89 | \( 1 + (-2.85e4 + 8.78e4i)T + (-4.51e9 - 3.28e9i)T^{2} \) |
| 97 | \( 1 + (-3.62e4 - 2.63e4i)T + (2.65e9 + 8.16e9i)T^{2} \) |
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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
−15.35087132354774061855826996394, −14.13521373918553197488687919057, −13.46529997753305730489556171601, −11.33846042112634878462992250244, −9.824751204171177981153067448697, −9.065664655300053416425886145547, −7.979606952868081290402584077640, −6.51038661125965385997290697140, −4.10958126525429101579226241909, −3.21880542422184976059764905710,
0.70157159792695740242452349271, 2.67306716347028693411647041355, 3.81793452542748929868586219897, 6.84097654593154728414473857499, 8.020442233818711865573846695050, 8.953005456102568729522074655863, 10.14590022639183153657752967026, 12.40680879152203637309144036846, 12.54279774127198499887973084286, 13.64577773792059447724509561774
