| 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
−15.52688587677357559675193452180, −15.11960580112582760821866815922, −13.74239563609383769312156962907, −12.33116479928357910333391925837, −10.77511545052553189305136835040, −9.165241788425083006037033302674, −8.269110398158592712480519916902, −6.92911759261155075052751085436, −4.57244115194103128593690191636, −2.72791146168742177279360569474,
1.85739959038178807670111483872, 3.21042019321762285239794869668, 6.26660297729030817528191430240, 7.67554782101716888726941961127, 8.986503279094446357189813566597, 10.52271649703933152850753600785, 11.70576973833210751888811435425, 13.01360739324205629372183580751, 14.37043467573392002369641318123, 15.00892921353941818035932188962
