L(s) = 1 | + (−0.670 + 0.562i)3-s + (−12.4 − 4.54i)5-s + (41.6 − 72.1i)7-s + (−42.0 + 238. i)9-s + (−226. − 392. i)11-s + (−370. − 311. i)13-s + (10.9 − 3.98i)15-s + (−138. − 784. i)17-s + (−2.60 − 1.57e3i)19-s + (12.6 + 71.8i)21-s + (−3.26e3 + 1.18e3i)23-s + (−2.25e3 − 1.89e3i)25-s + (−212. − 367. i)27-s + (1.08e3 − 6.14e3i)29-s + (816. − 1.41e3i)31-s + ⋯ |
L(s) = 1 | + (−0.0430 + 0.0361i)3-s + (−0.223 − 0.0813i)5-s + (0.321 − 0.556i)7-s + (−0.173 + 0.981i)9-s + (−0.564 − 0.978i)11-s + (−0.608 − 0.510i)13-s + (0.0125 − 0.00456i)15-s + (−0.116 − 0.658i)17-s + (−0.00165 − 0.999i)19-s + (0.00627 + 0.0355i)21-s + (−1.28 + 0.468i)23-s + (−0.722 − 0.606i)25-s + (−0.0560 − 0.0971i)27-s + (0.239 − 1.35i)29-s + (0.152 − 0.264i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.629 + 0.777i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.629 + 0.777i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.354282 - 0.742763i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.354282 - 0.742763i\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 + (2.60 + 1.57e3i)T \) |
good | 3 | \( 1 + (0.670 - 0.562i)T + (42.1 - 239. i)T^{2} \) |
| 5 | \( 1 + (12.4 + 4.54i)T + (2.39e3 + 2.00e3i)T^{2} \) |
| 7 | \( 1 + (-41.6 + 72.1i)T + (-8.40e3 - 1.45e4i)T^{2} \) |
| 11 | \( 1 + (226. + 392. i)T + (-8.05e4 + 1.39e5i)T^{2} \) |
| 13 | \( 1 + (370. + 311. i)T + (6.44e4 + 3.65e5i)T^{2} \) |
| 17 | \( 1 + (138. + 784. i)T + (-1.33e6 + 4.85e5i)T^{2} \) |
| 23 | \( 1 + (3.26e3 - 1.18e3i)T + (4.93e6 - 4.13e6i)T^{2} \) |
| 29 | \( 1 + (-1.08e3 + 6.14e3i)T + (-1.92e7 - 7.01e6i)T^{2} \) |
| 31 | \( 1 + (-816. + 1.41e3i)T + (-1.43e7 - 2.47e7i)T^{2} \) |
| 37 | \( 1 + 4.16e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + (1.17e4 - 9.85e3i)T + (2.01e7 - 1.14e8i)T^{2} \) |
| 43 | \( 1 + (-1.98e4 - 7.21e3i)T + (1.12e8 + 9.44e7i)T^{2} \) |
| 47 | \( 1 + (-961. + 5.45e3i)T + (-2.15e8 - 7.84e7i)T^{2} \) |
| 53 | \( 1 + (-1.93e4 + 7.03e3i)T + (3.20e8 - 2.68e8i)T^{2} \) |
| 59 | \( 1 + (-6.80e3 - 3.86e4i)T + (-6.71e8 + 2.44e8i)T^{2} \) |
| 61 | \( 1 + (2.48e4 - 9.05e3i)T + (6.46e8 - 5.42e8i)T^{2} \) |
| 67 | \( 1 + (-725. + 4.11e3i)T + (-1.26e9 - 4.61e8i)T^{2} \) |
| 71 | \( 1 + (-3.52e4 - 1.28e4i)T + (1.38e9 + 1.15e9i)T^{2} \) |
| 73 | \( 1 + (-2.43e4 + 2.03e4i)T + (3.59e8 - 2.04e9i)T^{2} \) |
| 79 | \( 1 + (-2.64e4 + 2.22e4i)T + (5.34e8 - 3.03e9i)T^{2} \) |
| 83 | \( 1 + (2.05e3 - 3.55e3i)T + (-1.96e9 - 3.41e9i)T^{2} \) |
| 89 | \( 1 + (8.08e4 + 6.78e4i)T + (9.69e8 + 5.49e9i)T^{2} \) |
| 97 | \( 1 + (1.65e4 + 9.38e4i)T + (-8.06e9 + 2.93e9i)T^{2} \) |
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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
−13.47929246322291143204664839612, −11.87862995475906524642073299362, −10.94269775136352749107250025591, −9.922796215147623426156173061270, −8.275100032597687167076760213153, −7.51009701704537745231434686195, −5.71900416184213864859151254667, −4.43884667880636285562832551727, −2.57437535208153957560433619775, −0.34052154620096585340046808917,
1.99969873333225412032052791114, 3.88947767740541887118170355625, 5.46771960181158287839606151774, 6.87726048482456800270567104028, 8.181185151835493315111541653898, 9.421984713530027319600022953447, 10.54319569814747265512281352682, 12.12704337479367269459731167145, 12.36778538223960133216422417659, 14.12540269511019726459071256689