| L(s) = 1 | + (−6.40 − 6.40i)2-s − 7.09i·3-s − 173. i·4-s + (773. − 773. i)5-s + (−45.4 + 45.4i)6-s + 1.90e3·7-s + (−2.75e3 + 2.75e3i)8-s + 6.51e3·9-s − 9.91e3·10-s + 9.26e3i·11-s − 1.23e3·12-s + (1.91e4 − 1.91e4i)13-s + (−1.22e4 − 1.22e4i)14-s + (−5.48e3 − 5.48e3i)15-s − 9.24e3·16-s + (9.88e4 − 9.88e4i)17-s + ⋯ |
| L(s) = 1 | + (−0.400 − 0.400i)2-s − 0.0875i·3-s − 0.679i·4-s + (1.23 − 1.23i)5-s + (−0.0350 + 0.0350i)6-s + 0.793·7-s + (−0.672 + 0.672i)8-s + 0.992·9-s − 0.991·10-s + 0.632i·11-s − 0.0594·12-s + (0.670 − 0.670i)13-s + (−0.317 − 0.317i)14-s + (−0.108 − 0.108i)15-s − 0.141·16-s + (1.18 − 1.18i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.410 + 0.912i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.410 + 0.912i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(1.12364 - 1.73723i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.12364 - 1.73723i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 + (-5.18e5 - 1.80e6i)T \) |
| good | 2 | \( 1 + (6.40 + 6.40i)T + 256iT^{2} \) |
| 3 | \( 1 + 7.09iT - 6.56e3T^{2} \) |
| 5 | \( 1 + (-773. + 773. i)T - 3.90e5iT^{2} \) |
| 7 | \( 1 - 1.90e3T + 5.76e6T^{2} \) |
| 11 | \( 1 - 9.26e3iT - 2.14e8T^{2} \) |
| 13 | \( 1 + (-1.91e4 + 1.91e4i)T - 8.15e8iT^{2} \) |
| 17 | \( 1 + (-9.88e4 + 9.88e4i)T - 6.97e9iT^{2} \) |
| 19 | \( 1 + (5.80e4 - 5.80e4i)T - 1.69e10iT^{2} \) |
| 23 | \( 1 + (2.75e5 - 2.75e5i)T - 7.83e10iT^{2} \) |
| 29 | \( 1 + (6.59e5 + 6.59e5i)T + 5.00e11iT^{2} \) |
| 31 | \( 1 + (-7.17e4 - 7.17e4i)T + 8.52e11iT^{2} \) |
| 41 | \( 1 - 1.43e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 + (3.21e6 - 3.21e6i)T - 1.16e13iT^{2} \) |
| 47 | \( 1 + 2.40e5T + 2.38e13T^{2} \) |
| 53 | \( 1 - 5.79e6T + 6.22e13T^{2} \) |
| 59 | \( 1 + (7.35e6 - 7.35e6i)T - 1.46e14iT^{2} \) |
| 61 | \( 1 + (-3.89e6 - 3.89e6i)T + 1.91e14iT^{2} \) |
| 67 | \( 1 + 2.46e7iT - 4.06e14T^{2} \) |
| 71 | \( 1 - 3.02e7T + 6.45e14T^{2} \) |
| 73 | \( 1 - 4.79e6iT - 8.06e14T^{2} \) |
| 79 | \( 1 + (1.51e7 - 1.51e7i)T - 1.51e15iT^{2} \) |
| 83 | \( 1 + 7.94e7T + 2.25e15T^{2} \) |
| 89 | \( 1 + (2.32e6 + 2.32e6i)T + 3.93e15iT^{2} \) |
| 97 | \( 1 + (-3.39e7 + 3.39e7i)T - 7.83e15iT^{2} \) |
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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
−14.05829682473343849117946303503, −13.05174500720950901877410456688, −11.75434679177011958611720392417, −10.05193961378533069821751669003, −9.580886779411536396244691897186, −8.050387416241751915458133006784, −5.86167770857597218734075065565, −4.84404456658891783105537737211, −1.81507519121692161733106774339, −1.08596269265607797636335536587,
1.87638886649191235993977299610, 3.70299720857669117281240768401, 6.03939250038892822498419150516, 7.13756502396161650556195229139, 8.557183392858926672834608860526, 10.00177582358728353873860283552, 11.04411400332926232924241853186, 12.72307694737740414426239593114, 13.96470835826576522588808658433, 14.92246712357416177713528147563
