| L(s) = 1 | + (−0.533 − 1.99i)2-s + (10.4 − 6.01i)3-s + (10.1 − 5.87i)4-s + (−3.50 + 13.0i)5-s + (−17.5 − 17.5i)6-s + (9.24 + 16.0i)7-s + (−40.4 − 40.4i)8-s + (31.7 − 55.0i)9-s + 27.9·10-s − 79.2i·11-s + (70.5 − 122. i)12-s + (−13.0 + 48.7i)13-s + (26.9 − 26.9i)14-s + (42.0 + 157. i)15-s + (34.9 − 60.4i)16-s + (−164. + 44.0i)17-s + ⋯ |
| L(s) = 1 | + (−0.133 − 0.498i)2-s + (1.15 − 0.667i)3-s + (0.635 − 0.367i)4-s + (−0.140 + 0.522i)5-s + (−0.487 − 0.487i)6-s + (0.188 + 0.326i)7-s + (−0.632 − 0.632i)8-s + (0.392 − 0.679i)9-s + 0.279·10-s − 0.655i·11-s + (0.490 − 0.849i)12-s + (−0.0773 + 0.288i)13-s + (0.137 − 0.137i)14-s + (0.187 + 0.698i)15-s + (0.136 − 0.236i)16-s + (−0.568 + 0.152i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.374 + 0.927i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.374 + 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.68967 - 1.13975i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.68967 - 1.13975i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 + (-146. + 1.36e3i)T \) |
| good | 2 | \( 1 + (0.533 + 1.99i)T + (-13.8 + 8i)T^{2} \) |
| 3 | \( 1 + (-10.4 + 6.01i)T + (40.5 - 70.1i)T^{2} \) |
| 5 | \( 1 + (3.50 - 13.0i)T + (-541. - 312.5i)T^{2} \) |
| 7 | \( 1 + (-9.24 - 16.0i)T + (-1.20e3 + 2.07e3i)T^{2} \) |
| 11 | \( 1 + 79.2iT - 1.46e4T^{2} \) |
| 13 | \( 1 + (13.0 - 48.7i)T + (-2.47e4 - 1.42e4i)T^{2} \) |
| 17 | \( 1 + (164. - 44.0i)T + (7.23e4 - 4.17e4i)T^{2} \) |
| 19 | \( 1 + (155. - 579. i)T + (-1.12e5 - 6.51e4i)T^{2} \) |
| 23 | \( 1 + (-99.9 - 99.9i)T + 2.79e5iT^{2} \) |
| 29 | \( 1 + (-116. + 116. i)T - 7.07e5iT^{2} \) |
| 31 | \( 1 + (-293. + 293. i)T - 9.23e5iT^{2} \) |
| 41 | \( 1 + (1.10e3 - 640. i)T + (1.41e6 - 2.44e6i)T^{2} \) |
| 43 | \( 1 + (-1.77e3 - 1.77e3i)T + 3.41e6iT^{2} \) |
| 47 | \( 1 + 2.61e3T + 4.87e6T^{2} \) |
| 53 | \( 1 + (-1.40e3 + 2.43e3i)T + (-3.94e6 - 6.83e6i)T^{2} \) |
| 59 | \( 1 + (2.92e3 - 784. i)T + (1.04e7 - 6.05e6i)T^{2} \) |
| 61 | \( 1 + (4.35e3 + 1.16e3i)T + (1.19e7 + 6.92e6i)T^{2} \) |
| 67 | \( 1 + (2.92e3 - 1.68e3i)T + (1.00e7 - 1.74e7i)T^{2} \) |
| 71 | \( 1 + (2.58e3 + 4.48e3i)T + (-1.27e7 + 2.20e7i)T^{2} \) |
| 73 | \( 1 - 3.31e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 + (-2.84e3 + 1.06e4i)T + (-3.37e7 - 1.94e7i)T^{2} \) |
| 83 | \( 1 + (-6.59e3 + 1.14e4i)T + (-2.37e7 - 4.11e7i)T^{2} \) |
| 89 | \( 1 + (3.14e3 + 1.17e4i)T + (-5.43e7 + 3.13e7i)T^{2} \) |
| 97 | \( 1 + (-3.12e3 - 3.12e3i)T + 8.85e7iT^{2} \) |
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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
−15.00892921353941818035932188962, −14.37043467573392002369641318123, −13.01360739324205629372183580751, −11.70576973833210751888811435425, −10.52271649703933152850753600785, −8.986503279094446357189813566597, −7.67554782101716888726941961127, −6.26660297729030817528191430240, −3.21042019321762285239794869668, −1.85739959038178807670111483872,
2.72791146168742177279360569474, 4.57244115194103128593690191636, 6.92911759261155075052751085436, 8.269110398158592712480519916902, 9.165241788425083006037033302674, 10.77511545052553189305136835040, 12.33116479928357910333391925837, 13.74239563609383769312156962907, 15.11960580112582760821866815922, 15.52688587677357559675193452180
