| L(s) = 1 | + (0.627 + 2.34i)2-s + (−0.550 + 0.317i)3-s + (8.76 − 5.05i)4-s + (12.3 − 46.2i)5-s + (−1.08 − 1.08i)6-s + (32.9 + 57.0i)7-s + (44.7 + 44.7i)8-s + (−40.2 + 69.7i)9-s + 116.·10-s − 143. i·11-s + (−3.21 + 5.56i)12-s + (22.9 − 85.5i)13-s + (−112. + 112. i)14-s + (7.86 + 29.3i)15-s + (4.13 − 7.15i)16-s + (−70.1 + 18.8i)17-s + ⋯ |
| L(s) = 1 | + (0.156 + 0.585i)2-s + (−0.0611 + 0.0352i)3-s + (0.547 − 0.316i)4-s + (0.495 − 1.84i)5-s + (−0.0302 − 0.0302i)6-s + (0.671 + 1.16i)7-s + (0.699 + 0.699i)8-s + (−0.497 + 0.861i)9-s + 1.16·10-s − 1.18i·11-s + (−0.0223 + 0.0386i)12-s + (0.135 − 0.506i)13-s + (−0.576 + 0.576i)14-s + (0.0349 + 0.130i)15-s + (0.0161 − 0.0279i)16-s + (−0.242 + 0.0650i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0643i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.997 - 0.0643i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.90124 + 0.0612249i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.90124 + 0.0612249i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 + (1.24e3 + 567. i)T \) |
| good | 2 | \( 1 + (-0.627 - 2.34i)T + (-13.8 + 8i)T^{2} \) |
| 3 | \( 1 + (0.550 - 0.317i)T + (40.5 - 70.1i)T^{2} \) |
| 5 | \( 1 + (-12.3 + 46.2i)T + (-541. - 312.5i)T^{2} \) |
| 7 | \( 1 + (-32.9 - 57.0i)T + (-1.20e3 + 2.07e3i)T^{2} \) |
| 11 | \( 1 + 143. iT - 1.46e4T^{2} \) |
| 13 | \( 1 + (-22.9 + 85.5i)T + (-2.47e4 - 1.42e4i)T^{2} \) |
| 17 | \( 1 + (70.1 - 18.8i)T + (7.23e4 - 4.17e4i)T^{2} \) |
| 19 | \( 1 + (96.2 - 359. i)T + (-1.12e5 - 6.51e4i)T^{2} \) |
| 23 | \( 1 + (-215. - 215. i)T + 2.79e5iT^{2} \) |
| 29 | \( 1 + (623. - 623. i)T - 7.07e5iT^{2} \) |
| 31 | \( 1 + (902. - 902. i)T - 9.23e5iT^{2} \) |
| 41 | \( 1 + (44.9 - 25.9i)T + (1.41e6 - 2.44e6i)T^{2} \) |
| 43 | \( 1 + (370. + 370. i)T + 3.41e6iT^{2} \) |
| 47 | \( 1 - 1.10e3T + 4.87e6T^{2} \) |
| 53 | \( 1 + (943. - 1.63e3i)T + (-3.94e6 - 6.83e6i)T^{2} \) |
| 59 | \( 1 + (-3.36e3 + 901. i)T + (1.04e7 - 6.05e6i)T^{2} \) |
| 61 | \( 1 + (-6.21e3 - 1.66e3i)T + (1.19e7 + 6.92e6i)T^{2} \) |
| 67 | \( 1 + (-510. + 294. i)T + (1.00e7 - 1.74e7i)T^{2} \) |
| 71 | \( 1 + (3.34e3 + 5.79e3i)T + (-1.27e7 + 2.20e7i)T^{2} \) |
| 73 | \( 1 - 4.01e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 + (-589. + 2.19e3i)T + (-3.37e7 - 1.94e7i)T^{2} \) |
| 83 | \( 1 + (-4.32e3 + 7.49e3i)T + (-2.37e7 - 4.11e7i)T^{2} \) |
| 89 | \( 1 + (-2.84e3 - 1.06e4i)T + (-5.43e7 + 3.13e7i)T^{2} \) |
| 97 | \( 1 + (1.91e3 + 1.91e3i)T + 8.85e7iT^{2} \) |
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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
−15.95809766563323802708338818218, −14.53041908263775970037208869023, −13.37036903067871023127752902671, −12.08006148941463094931913225519, −10.84305529711308238168907387686, −8.841726235361013945124866177980, −8.153536890776694041892354969598, −5.54829558842391251231387046938, −5.38022461378129611460237398547, −1.75456790834670142390025023550,
2.24533734722194893516162705255, 3.84645009305962142031900245579, 6.68930365499075250666943443071, 7.29363585992218425364621727316, 9.853203012200717453267034787821, 10.92952387754976209601067510087, 11.54528285193576005652853944044, 13.26133052313811032402935013725, 14.46705647212445581145404391771, 15.23319677322399695854717471089
