| L(s) = 1 | + (−13.4 − 11.3i)3-s + (63.2 − 23.0i)5-s + (73.8 + 127. i)7-s + (11.6 + 65.9i)9-s + (48.0 − 83.1i)11-s + (19.6 − 16.4i)13-s + (−1.11e3 − 405. i)15-s + (388. − 2.20e3i)17-s + (887. − 1.29e3i)19-s + (451. − 2.55e3i)21-s + (−890. − 324. i)23-s + (1.07e3 − 901. i)25-s + (−1.54e3 + 2.68e3i)27-s + (−1.21e3 − 6.89e3i)29-s + (1.78e3 + 3.09e3i)31-s + ⋯ |
| L(s) = 1 | + (−0.865 − 0.726i)3-s + (1.13 − 0.411i)5-s + (0.569 + 0.986i)7-s + (0.0478 + 0.271i)9-s + (0.119 − 0.207i)11-s + (0.0321 − 0.0270i)13-s + (−1.27 − 0.465i)15-s + (0.325 − 1.84i)17-s + (0.564 − 0.825i)19-s + (0.223 − 1.26i)21-s + (−0.350 − 0.127i)23-s + (0.343 − 0.288i)25-s + (−0.409 + 0.708i)27-s + (−0.268 − 1.52i)29-s + (0.333 + 0.577i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0826 + 0.996i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.0826 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.16745 - 1.07464i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.16745 - 1.07464i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (-887. + 1.29e3i)T \) |
| good | 3 | \( 1 + (13.4 + 11.3i)T + (42.1 + 239. i)T^{2} \) |
| 5 | \( 1 + (-63.2 + 23.0i)T + (2.39e3 - 2.00e3i)T^{2} \) |
| 7 | \( 1 + (-73.8 - 127. i)T + (-8.40e3 + 1.45e4i)T^{2} \) |
| 11 | \( 1 + (-48.0 + 83.1i)T + (-8.05e4 - 1.39e5i)T^{2} \) |
| 13 | \( 1 + (-19.6 + 16.4i)T + (6.44e4 - 3.65e5i)T^{2} \) |
| 17 | \( 1 + (-388. + 2.20e3i)T + (-1.33e6 - 4.85e5i)T^{2} \) |
| 23 | \( 1 + (890. + 324. i)T + (4.93e6 + 4.13e6i)T^{2} \) |
| 29 | \( 1 + (1.21e3 + 6.89e3i)T + (-1.92e7 + 7.01e6i)T^{2} \) |
| 31 | \( 1 + (-1.78e3 - 3.09e3i)T + (-1.43e7 + 2.47e7i)T^{2} \) |
| 37 | \( 1 - 506.T + 6.93e7T^{2} \) |
| 41 | \( 1 + (5.82e3 + 4.89e3i)T + (2.01e7 + 1.14e8i)T^{2} \) |
| 43 | \( 1 + (-8.74e3 + 3.18e3i)T + (1.12e8 - 9.44e7i)T^{2} \) |
| 47 | \( 1 + (2.18e3 + 1.23e4i)T + (-2.15e8 + 7.84e7i)T^{2} \) |
| 53 | \( 1 + (-5.98e3 - 2.17e3i)T + (3.20e8 + 2.68e8i)T^{2} \) |
| 59 | \( 1 + (5.91e3 - 3.35e4i)T + (-6.71e8 - 2.44e8i)T^{2} \) |
| 61 | \( 1 + (-4.94e4 - 1.79e4i)T + (6.46e8 + 5.42e8i)T^{2} \) |
| 67 | \( 1 + (-6.10e3 - 3.46e4i)T + (-1.26e9 + 4.61e8i)T^{2} \) |
| 71 | \( 1 + (-4.09e4 + 1.49e4i)T + (1.38e9 - 1.15e9i)T^{2} \) |
| 73 | \( 1 + (-3.34e3 - 2.80e3i)T + (3.59e8 + 2.04e9i)T^{2} \) |
| 79 | \( 1 + (7.43e4 + 6.23e4i)T + (5.34e8 + 3.03e9i)T^{2} \) |
| 83 | \( 1 + (-2.37e4 - 4.11e4i)T + (-1.96e9 + 3.41e9i)T^{2} \) |
| 89 | \( 1 + (5.23e4 - 4.39e4i)T + (9.69e8 - 5.49e9i)T^{2} \) |
| 97 | \( 1 + (-7.86e3 + 4.45e4i)T + (-8.06e9 - 2.93e9i)T^{2} \) |
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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
−13.24581632763251514011459788544, −11.98248732044660289805287638540, −11.50955081605914121956730450296, −9.787543494777537734035577367921, −8.812453607920836999752021287002, −7.15277162533495208655764635430, −5.83547111683063755022819691079, −5.17186952822922271920207348777, −2.35120468156013316074728967327, −0.827819703451814914910199309232,
1.59282681429553070908170451827, 3.95012807815847685450555864641, 5.35079166039612967422627348505, 6.36575747325411216520136693475, 7.961237548149365131195497759269, 9.801049203554892947248634524538, 10.43147123002174529154384607045, 11.20457083917120248704494246526, 12.70158746964245818114133366003, 13.96709773511048178285803837244
