| L(s) = 1 | + (−7.74 + 4.58i)3-s + (−2.39 − 6.57i)5-s + (−4.62 − 26.2i)7-s + (38.8 − 71.0i)9-s + (−16.9 + 46.5i)11-s + (94.6 + 79.4i)13-s + (48.7 + 39.9i)15-s + (285. + 164. i)17-s + (116. + 201. i)19-s + (156. + 181. i)21-s + (392. + 69.1i)23-s + (441. − 370. i)25-s + (24.9 + 728. i)27-s + (555. + 662. i)29-s + (−8.30 + 47.1i)31-s + ⋯ |
| L(s) = 1 | + (−0.860 + 0.509i)3-s + (−0.0957 − 0.263i)5-s + (−0.0943 − 0.534i)7-s + (0.480 − 0.877i)9-s + (−0.139 + 0.384i)11-s + (0.560 + 0.470i)13-s + (0.216 + 0.177i)15-s + (0.986 + 0.569i)17-s + (0.322 + 0.558i)19-s + (0.353 + 0.412i)21-s + (0.741 + 0.130i)23-s + (0.705 − 0.592i)25-s + (0.0341 + 0.999i)27-s + (0.660 + 0.787i)29-s + (−0.00864 + 0.0490i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.904 - 0.427i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.904 - 0.427i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.25455 + 0.281477i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.25455 + 0.281477i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (7.74 - 4.58i)T \) |
| good | 5 | \( 1 + (2.39 + 6.57i)T + (-478. + 401. i)T^{2} \) |
| 7 | \( 1 + (4.62 + 26.2i)T + (-2.25e3 + 821. i)T^{2} \) |
| 11 | \( 1 + (16.9 - 46.5i)T + (-1.12e4 - 9.41e3i)T^{2} \) |
| 13 | \( 1 + (-94.6 - 79.4i)T + (4.95e3 + 2.81e4i)T^{2} \) |
| 17 | \( 1 + (-285. - 164. i)T + (4.17e4 + 7.23e4i)T^{2} \) |
| 19 | \( 1 + (-116. - 201. i)T + (-6.51e4 + 1.12e5i)T^{2} \) |
| 23 | \( 1 + (-392. - 69.1i)T + (2.62e5 + 9.57e4i)T^{2} \) |
| 29 | \( 1 + (-555. - 662. i)T + (-1.22e5 + 6.96e5i)T^{2} \) |
| 31 | \( 1 + (8.30 - 47.1i)T + (-8.67e5 - 3.15e5i)T^{2} \) |
| 37 | \( 1 + (-822. + 1.42e3i)T + (-9.37e5 - 1.62e6i)T^{2} \) |
| 41 | \( 1 + (-242. + 289. i)T + (-4.90e5 - 2.78e6i)T^{2} \) |
| 43 | \( 1 + (807. + 293. i)T + (2.61e6 + 2.19e6i)T^{2} \) |
| 47 | \( 1 + (579. - 102. i)T + (4.58e6 - 1.66e6i)T^{2} \) |
| 53 | \( 1 - 1.45e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 + (-1.90e3 - 5.23e3i)T + (-9.28e6 + 7.78e6i)T^{2} \) |
| 61 | \( 1 + (744. + 4.22e3i)T + (-1.30e7 + 4.73e6i)T^{2} \) |
| 67 | \( 1 + (2.80e3 + 2.35e3i)T + (3.49e6 + 1.98e7i)T^{2} \) |
| 71 | \( 1 + (-68.5 - 39.5i)T + (1.27e7 + 2.20e7i)T^{2} \) |
| 73 | \( 1 + (1.30e3 + 2.25e3i)T + (-1.41e7 + 2.45e7i)T^{2} \) |
| 79 | \( 1 + (-4.20e3 + 3.52e3i)T + (6.76e6 - 3.83e7i)T^{2} \) |
| 83 | \( 1 + (-2.37e3 - 2.82e3i)T + (-8.24e6 + 4.67e7i)T^{2} \) |
| 89 | \( 1 + (3.20e3 - 1.85e3i)T + (3.13e7 - 5.43e7i)T^{2} \) |
| 97 | \( 1 + (1.63e3 + 595. i)T + (6.78e7 + 5.69e7i)T^{2} \) |
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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
−12.80268021282959466194650078218, −12.02751953060132304141680667644, −10.83962950438894815220732400151, −10.09782090597852195287342121247, −8.884420374183392141142224809733, −7.36424228829224682469793972875, −6.13176794346063555290791748654, −4.87231624779704129866709773026, −3.66193389932885121711572522907, −1.04678029855968041489117173429,
0.899259434426999293522116132881, 2.94259666154902292824162436500, 5.00874067187018676129081232585, 6.04654554696319179301956313319, 7.20414317888073169265960389739, 8.394745461353559779404120565483, 9.883207768497967068947296861608, 11.04567320482916155836149702196, 11.78388310769138709297523636416, 12.85029305336174122307809032628
