| L(s) = 1 | + (7.44 + 7.44i)2-s + 17.9i·3-s + 46.7i·4-s + (−84.4 + 84.4i)5-s + (−133. + 133. i)6-s − 514.·7-s + (128. − 128. i)8-s + 405.·9-s − 1.25e3·10-s + 1.75e3i·11-s − 840.·12-s + (399. − 399. i)13-s + (−3.82e3 − 3.82e3i)14-s + (−1.52e3 − 1.52e3i)15-s + 4.90e3·16-s + (−6.19e3 + 6.19e3i)17-s + ⋯ |
| L(s) = 1 | + (0.930 + 0.930i)2-s + 0.666i·3-s + 0.730i·4-s + (−0.675 + 0.675i)5-s + (−0.619 + 0.619i)6-s − 1.50·7-s + (0.251 − 0.251i)8-s + 0.555·9-s − 1.25·10-s + 1.31i·11-s − 0.486·12-s + (0.181 − 0.181i)13-s + (−1.39 − 1.39i)14-s + (−0.450 − 0.450i)15-s + 1.19·16-s + (−1.26 + 1.26i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 - 0.119i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.992 - 0.119i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.116305 + 1.93208i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.116305 + 1.93208i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 + (-4.22e4 + 2.78e4i)T \) |
| good | 2 | \( 1 + (-7.44 - 7.44i)T + 64iT^{2} \) |
| 3 | \( 1 - 17.9iT - 729T^{2} \) |
| 5 | \( 1 + (84.4 - 84.4i)T - 1.56e4iT^{2} \) |
| 7 | \( 1 + 514.T + 1.17e5T^{2} \) |
| 11 | \( 1 - 1.75e3iT - 1.77e6T^{2} \) |
| 13 | \( 1 + (-399. + 399. i)T - 4.82e6iT^{2} \) |
| 17 | \( 1 + (6.19e3 - 6.19e3i)T - 2.41e7iT^{2} \) |
| 19 | \( 1 + (-7.52e3 + 7.52e3i)T - 4.70e7iT^{2} \) |
| 23 | \( 1 + (2.18e3 - 2.18e3i)T - 1.48e8iT^{2} \) |
| 29 | \( 1 + (-681. - 681. i)T + 5.94e8iT^{2} \) |
| 31 | \( 1 + (-3.73e4 - 3.73e4i)T + 8.87e8iT^{2} \) |
| 41 | \( 1 + 1.44e3iT - 4.75e9T^{2} \) |
| 43 | \( 1 + (-8.48e4 + 8.48e4i)T - 6.32e9iT^{2} \) |
| 47 | \( 1 + 1.41e5T + 1.07e10T^{2} \) |
| 53 | \( 1 + 6.83e4T + 2.21e10T^{2} \) |
| 59 | \( 1 + (-2.28e4 + 2.28e4i)T - 4.21e10iT^{2} \) |
| 61 | \( 1 + (2.16e5 + 2.16e5i)T + 5.15e10iT^{2} \) |
| 67 | \( 1 - 3.00e5iT - 9.04e10T^{2} \) |
| 71 | \( 1 - 4.45e4T + 1.28e11T^{2} \) |
| 73 | \( 1 + 3.34e5iT - 1.51e11T^{2} \) |
| 79 | \( 1 + (-3.39e5 + 3.39e5i)T - 2.43e11iT^{2} \) |
| 83 | \( 1 - 6.80e4T + 3.26e11T^{2} \) |
| 89 | \( 1 + (-2.41e5 - 2.41e5i)T + 4.96e11iT^{2} \) |
| 97 | \( 1 + (-2.31e5 + 2.31e5i)T - 8.32e11iT^{2} \) |
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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
−15.60235552229449414871184020224, −14.96014252125441452017415995322, −13.39143464554680673827757833150, −12.52780839160683464495718992598, −10.61229933567027392042126109367, −9.553290266638668053628492995986, −7.27453878569829710024460927530, −6.50102341028194144412598102265, −4.62024959183720465543720155364, −3.48504436722218654504498596418,
0.74874558749505389536193881168, 2.93931078110438109120185646805, 4.29984447933087558359328601279, 6.24069955198815773131449508359, 7.957990904041121946394318781114, 9.689822224347889259100064983979, 11.40878880929561878128163067992, 12.25593373664910778154852164823, 13.22799931792901500414746712467, 13.75562029209404585386447761981
