| L(s) = 1 | + (1.32 + 2.29i)2-s + (−4.04 − 3.25i)3-s + (0.476 − 0.824i)4-s + 7.35·5-s + (2.10 − 13.6i)6-s + (16.6 − 8.08i)7-s + 23.7·8-s + (5.80 + 26.3i)9-s + (9.76 + 16.9i)10-s + 11.3·11-s + (−4.61 + 1.78i)12-s + (−30.2 − 52.4i)13-s + (40.7 + 27.5i)14-s + (−29.7 − 23.9i)15-s + (27.7 + 48.0i)16-s + (−5.29 − 9.16i)17-s + ⋯ |
| L(s) = 1 | + (0.469 + 0.812i)2-s + (−0.779 − 0.626i)3-s + (0.0595 − 0.103i)4-s + 0.657·5-s + (0.143 − 0.927i)6-s + (0.899 − 0.436i)7-s + 1.05·8-s + (0.214 + 0.976i)9-s + (0.308 + 0.534i)10-s + 0.310·11-s + (−0.110 + 0.0430i)12-s + (−0.646 − 1.11i)13-s + (0.777 + 0.526i)14-s + (−0.512 − 0.412i)15-s + (0.433 + 0.750i)16-s + (−0.0755 − 0.130i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0338i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0338i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.79888 + 0.0304346i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.79888 + 0.0304346i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (4.04 + 3.25i)T \) |
| 7 | \( 1 + (-16.6 + 8.08i)T \) |
| good | 2 | \( 1 + (-1.32 - 2.29i)T + (-4 + 6.92i)T^{2} \) |
| 5 | \( 1 - 7.35T + 125T^{2} \) |
| 11 | \( 1 - 11.3T + 1.33e3T^{2} \) |
| 13 | \( 1 + (30.2 + 52.4i)T + (-1.09e3 + 1.90e3i)T^{2} \) |
| 17 | \( 1 + (5.29 + 9.16i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (41.6 - 72.1i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 - 87.6T + 1.21e4T^{2} \) |
| 29 | \( 1 + (67.6 - 117. i)T + (-1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 + (26.7 - 46.3i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (149. - 259. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + (194. + 336. i)T + (-3.44e4 + 5.96e4i)T^{2} \) |
| 43 | \( 1 + (215. - 373. i)T + (-3.97e4 - 6.88e4i)T^{2} \) |
| 47 | \( 1 + (-255. - 442. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-116. - 201. i)T + (-7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (31.1 - 54.0i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (170. + 294. i)T + (-1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-274. + 475. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 - 505.T + 3.57e5T^{2} \) |
| 73 | \( 1 + (99.1 + 171. i)T + (-1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (588. + 1.02e3i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + (722. - 1.25e3i)T + (-2.85e5 - 4.95e5i)T^{2} \) |
| 89 | \( 1 + (507. - 878. i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + (-551. + 954. i)T + (-4.56e5 - 7.90e5i)T^{2} \) |
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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
−14.40041970572144492879876739884, −13.57447195200531418811277678984, −12.46929745387873921157363878112, −11.03004090466496117922966573684, −10.19236719744668827669956558140, −7.999258229117291490707265222163, −6.98553356098485978765342967148, −5.76457411390471709697456365922, −4.87324545282759173744881127628, −1.54983872757207716544448671567,
2.04577248773726718661157198814, 4.16350799395386093549021589269, 5.28644103683794268572590851271, 6.96079744646188097506322704961, 8.957819299907322717544484520273, 10.20776804203533137782976446156, 11.38656652232027060109471253657, 11.84674721493518550892704745505, 13.13208295333876358778583547341, 14.34860710920661506736915340983
