| L(s) = 1 | + (3.37 − 3.37i)2-s + 108. i·3-s + 233. i·4-s + (154. + 154. i)5-s + (367. + 367. i)6-s + 3.84e3·7-s + (1.65e3 + 1.65e3i)8-s − 5.28e3·9-s + 1.04e3·10-s + 1.49e4i·11-s − 2.53e4·12-s + (−1.07e4 − 1.07e4i)13-s + (1.29e4 − 1.29e4i)14-s + (−1.68e4 + 1.68e4i)15-s − 4.85e4·16-s + (−8.60e3 − 8.60e3i)17-s + ⋯ |
| L(s) = 1 | + (0.210 − 0.210i)2-s + 1.34i·3-s + 0.911i·4-s + (0.247 + 0.247i)5-s + (0.283 + 0.283i)6-s + 1.59·7-s + (0.402 + 0.402i)8-s − 0.805·9-s + 0.104·10-s + 1.01i·11-s − 1.22·12-s + (−0.377 − 0.377i)13-s + (0.337 − 0.337i)14-s + (−0.333 + 0.333i)15-s − 0.741·16-s + (−0.103 − 0.103i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.620 - 0.784i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.620 - 0.784i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(1.04048 + 2.15039i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.04048 + 2.15039i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 + (-6.24e4 + 1.87e6i)T \) |
| good | 2 | \( 1 + (-3.37 + 3.37i)T - 256iT^{2} \) |
| 3 | \( 1 - 108. iT - 6.56e3T^{2} \) |
| 5 | \( 1 + (-154. - 154. i)T + 3.90e5iT^{2} \) |
| 7 | \( 1 - 3.84e3T + 5.76e6T^{2} \) |
| 11 | \( 1 - 1.49e4iT - 2.14e8T^{2} \) |
| 13 | \( 1 + (1.07e4 + 1.07e4i)T + 8.15e8iT^{2} \) |
| 17 | \( 1 + (8.60e3 + 8.60e3i)T + 6.97e9iT^{2} \) |
| 19 | \( 1 + (4.53e4 + 4.53e4i)T + 1.69e10iT^{2} \) |
| 23 | \( 1 + (-5.69e4 - 5.69e4i)T + 7.83e10iT^{2} \) |
| 29 | \( 1 + (-9.00e5 + 9.00e5i)T - 5.00e11iT^{2} \) |
| 31 | \( 1 + (8.92e5 - 8.92e5i)T - 8.52e11iT^{2} \) |
| 41 | \( 1 + 3.05e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 + (-3.09e6 - 3.09e6i)T + 1.16e13iT^{2} \) |
| 47 | \( 1 + 9.76e5T + 2.38e13T^{2} \) |
| 53 | \( 1 - 5.94e6T + 6.22e13T^{2} \) |
| 59 | \( 1 + (3.18e6 + 3.18e6i)T + 1.46e14iT^{2} \) |
| 61 | \( 1 + (-1.41e6 + 1.41e6i)T - 1.91e14iT^{2} \) |
| 67 | \( 1 - 1.38e7iT - 4.06e14T^{2} \) |
| 71 | \( 1 + 4.14e7T + 6.45e14T^{2} \) |
| 73 | \( 1 - 3.14e7iT - 8.06e14T^{2} \) |
| 79 | \( 1 + (-3.53e7 - 3.53e7i)T + 1.51e15iT^{2} \) |
| 83 | \( 1 - 5.42e7T + 2.25e15T^{2} \) |
| 89 | \( 1 + (-3.24e7 + 3.24e7i)T - 3.93e15iT^{2} \) |
| 97 | \( 1 + (-7.46e7 - 7.46e7i)T + 7.83e15iT^{2} \) |
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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
−15.00506426263113424170830882850, −14.10787974604258882971637503888, −12.43495459785609703418944698405, −11.24549364082052138558859368332, −10.26428818334749802111207849553, −8.784153924905226069419509534438, −7.48410111985543438651043464595, −4.96497235641589966372784944383, −4.18134750199102314637917988073, −2.34550551012413761132843463794,
0.977208520936398958358880599779, 1.87962736978080929543748559735, 4.89890748333196783187083962905, 6.15798495337371940749381658280, 7.51374557621699666280611089034, 8.781901569140127596941402191348, 10.72503233739324600064174605708, 11.79919065645144242603744363225, 13.25707081565604094022412902904, 14.10645463232608256720813134582
