L(s) = 1 | + (4.61 + 14.8i)3-s + (−31.2 − 46.3i)5-s + (−91.3 + 91.3i)7-s + (−200. + 137. i)9-s − 99.2i·11-s + (−452. − 452. i)13-s + (545. − 679. i)15-s + (−1.26e3 − 1.26e3i)17-s + 912. i·19-s + (−1.78e3 − 939. i)21-s + (−1.74e3 + 1.74e3i)23-s + (−1.16e3 + 2.89e3i)25-s + (−2.97e3 − 2.35e3i)27-s − 299.·29-s + 2.59e3·31-s + ⋯ |
L(s) = 1 | + (0.295 + 0.955i)3-s + (−0.559 − 0.828i)5-s + (−0.705 + 0.705i)7-s + (−0.824 + 0.565i)9-s − 0.247i·11-s + (−0.742 − 0.742i)13-s + (0.625 − 0.780i)15-s + (−1.06 − 1.06i)17-s + 0.580i·19-s + (−0.882 − 0.464i)21-s + (−0.688 + 0.688i)23-s + (−0.373 + 0.927i)25-s + (−0.784 − 0.620i)27-s − 0.0661·29-s + 0.484·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.942 + 0.334i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.942 + 0.334i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.0276879 - 0.160758i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0276879 - 0.160758i\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-4.61 - 14.8i)T \) |
| 5 | \( 1 + (31.2 + 46.3i)T \) |
good | 7 | \( 1 + (91.3 - 91.3i)T - 1.68e4iT^{2} \) |
| 11 | \( 1 + 99.2iT - 1.61e5T^{2} \) |
| 13 | \( 1 + (452. + 452. i)T + 3.71e5iT^{2} \) |
| 17 | \( 1 + (1.26e3 + 1.26e3i)T + 1.41e6iT^{2} \) |
| 19 | \( 1 - 912. iT - 2.47e6T^{2} \) |
| 23 | \( 1 + (1.74e3 - 1.74e3i)T - 6.43e6iT^{2} \) |
| 29 | \( 1 + 299.T + 2.05e7T^{2} \) |
| 31 | \( 1 - 2.59e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + (-5.91e3 + 5.91e3i)T - 6.93e7iT^{2} \) |
| 41 | \( 1 - 1.94e4iT - 1.15e8T^{2} \) |
| 43 | \( 1 + (8.60e3 + 8.60e3i)T + 1.47e8iT^{2} \) |
| 47 | \( 1 + (-1.73e4 - 1.73e4i)T + 2.29e8iT^{2} \) |
| 53 | \( 1 + (1.94e4 - 1.94e4i)T - 4.18e8iT^{2} \) |
| 59 | \( 1 + 220.T + 7.14e8T^{2} \) |
| 61 | \( 1 - 4.39e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + (3.86e4 - 3.86e4i)T - 1.35e9iT^{2} \) |
| 71 | \( 1 + 6.18e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 + (1.40e4 + 1.40e4i)T + 2.07e9iT^{2} \) |
| 79 | \( 1 + 6.48e4iT - 3.07e9T^{2} \) |
| 83 | \( 1 + (5.90e3 - 5.90e3i)T - 3.93e9iT^{2} \) |
| 89 | \( 1 + 1.25e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + (7.06e4 - 7.06e4i)T - 8.58e9iT^{2} \) |
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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
−15.02061338410048662103562070196, −13.59485708201374921239433099654, −12.40128932819558025187784404317, −11.34068092777403403836791484638, −9.817439928607655520517297423442, −9.020269044761666655498964030420, −7.83204054083171251550205568852, −5.71339477570731937788715477402, −4.45608992017659051052174447939, −2.89142982054699192583271354109,
0.06966336966313020284734351226, 2.36609088244197290938468359100, 3.98350554347688604335370165004, 6.51921145483580625195900505970, 7.11068562950670200202322189300, 8.451798794234376452569965416674, 10.02815271909952410404428741380, 11.31003366711749021945315943712, 12.42313931841766850789269364460, 13.48724167887929523472587645909