| L(s) = 1 | + (−3.72 + 6.44i)2-s + (5.98 + 10.3i)3-s + (−11.7 − 20.3i)4-s + (37.3 + 64.7i)5-s − 89.0·6-s + (68.5 + 118. i)7-s − 63.6·8-s + (49.9 − 86.4i)9-s − 556.·10-s + 109.·11-s + (140. − 242. i)12-s + (−500. − 867. i)13-s − 1.02e3·14-s + (−447. + 774. i)15-s + (612. − 1.06e3i)16-s + (−451. + 782. i)17-s + ⋯ |
| L(s) = 1 | + (−0.658 + 1.13i)2-s + (0.383 + 0.664i)3-s + (−0.366 − 0.634i)4-s + (0.668 + 1.15i)5-s − 1.01·6-s + (0.528 + 0.916i)7-s − 0.351·8-s + (0.205 − 0.355i)9-s − 1.76·10-s + 0.273·11-s + (0.281 − 0.487i)12-s + (−0.821 − 1.42i)13-s − 1.39·14-s + (−0.513 + 0.888i)15-s + (0.597 − 1.03i)16-s + (−0.378 + 0.656i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0260i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0260i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.0179852 + 1.37950i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0179852 + 1.37950i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 + (-2.85e3 - 7.82e3i)T \) |
| good | 2 | \( 1 + (3.72 - 6.44i)T + (-16 - 27.7i)T^{2} \) |
| 3 | \( 1 + (-5.98 - 10.3i)T + (-121.5 + 210. i)T^{2} \) |
| 5 | \( 1 + (-37.3 - 64.7i)T + (-1.56e3 + 2.70e3i)T^{2} \) |
| 7 | \( 1 + (-68.5 - 118. i)T + (-8.40e3 + 1.45e4i)T^{2} \) |
| 11 | \( 1 - 109.T + 1.61e5T^{2} \) |
| 13 | \( 1 + (500. + 867. i)T + (-1.85e5 + 3.21e5i)T^{2} \) |
| 17 | \( 1 + (451. - 782. i)T + (-7.09e5 - 1.22e6i)T^{2} \) |
| 19 | \( 1 + (-425. - 736. i)T + (-1.23e6 + 2.14e6i)T^{2} \) |
| 23 | \( 1 - 1.60e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 6.17e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 8.27e3T + 2.86e7T^{2} \) |
| 41 | \( 1 + (3.99e3 + 6.91e3i)T + (-5.79e7 + 1.00e8i)T^{2} \) |
| 43 | \( 1 + 3.86e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.72e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + (1.42e4 - 2.46e4i)T + (-2.09e8 - 3.62e8i)T^{2} \) |
| 59 | \( 1 + (-1.35e4 + 2.35e4i)T + (-3.57e8 - 6.19e8i)T^{2} \) |
| 61 | \( 1 + (1.12e4 + 1.94e4i)T + (-4.22e8 + 7.31e8i)T^{2} \) |
| 67 | \( 1 + (-2.72e4 - 4.71e4i)T + (-6.75e8 + 1.16e9i)T^{2} \) |
| 71 | \( 1 + (299. + 518. i)T + (-9.02e8 + 1.56e9i)T^{2} \) |
| 73 | \( 1 - 7.98e3T + 2.07e9T^{2} \) |
| 79 | \( 1 + (6.65e3 + 1.15e4i)T + (-1.53e9 + 2.66e9i)T^{2} \) |
| 83 | \( 1 + (-560. + 970. i)T + (-1.96e9 - 3.41e9i)T^{2} \) |
| 89 | \( 1 + (4.07e4 - 7.06e4i)T + (-2.79e9 - 4.83e9i)T^{2} \) |
| 97 | \( 1 + 1.06e5T + 8.58e9T^{2} \) |
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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
−15.60805001228859061179848210221, −15.00395423035984128635177799106, −14.41231074241778914930537400330, −12.34280046563001519641311860098, −10.50919544039693924265933419828, −9.481411541722360526689828042016, −8.289536184135038896125037546884, −6.83579299996658927794778789256, −5.55332704361319237741501180252, −2.92330803740810421236419557713,
1.02778208004052291149870132202, 2.09749675946845700211834705431, 4.69959170932300880826192062155, 7.13608889258309264544040933981, 8.762036478227360597808763856005, 9.634972292689241045105417260130, 11.05579750741703344772710605306, 12.27246348887241484948823413285, 13.33766775574281797841502473948, 14.27662397880905708106751698018
