| L(s) = 1 | + (3.64 − 20.6i)3-s + (83.5 + 70.1i)5-s + (58.2 + 100. i)7-s + (−185. − 67.4i)9-s + (−313. + 542. i)11-s + (117. + 665. i)13-s + (1.75e3 − 1.47e3i)15-s + (735. − 267. i)17-s + (−229. − 1.55e3i)19-s + (2.29e3 − 836. i)21-s + (1.88e3 − 1.58e3i)23-s + (1.52e3 + 8.63e3i)25-s + (481. − 834. i)27-s + (−1.42e3 − 517. i)29-s + (−3.99e3 − 6.91e3i)31-s + ⋯ |
| L(s) = 1 | + (0.233 − 1.32i)3-s + (1.49 + 1.25i)5-s + (0.449 + 0.778i)7-s + (−0.762 − 0.277i)9-s + (−0.780 + 1.35i)11-s + (0.192 + 1.09i)13-s + (2.01 − 1.68i)15-s + (0.616 − 0.224i)17-s + (−0.145 − 0.989i)19-s + (1.13 − 0.413i)21-s + (0.742 − 0.622i)23-s + (0.487 + 2.76i)25-s + (0.127 − 0.220i)27-s + (−0.313 − 0.114i)29-s + (−0.746 − 1.29i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 - 0.184i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.982 - 0.184i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.44300 + 0.226831i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.44300 + 0.226831i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (229. + 1.55e3i)T \) |
| good | 3 | \( 1 + (-3.64 + 20.6i)T + (-228. - 83.1i)T^{2} \) |
| 5 | \( 1 + (-83.5 - 70.1i)T + (542. + 3.07e3i)T^{2} \) |
| 7 | \( 1 + (-58.2 - 100. i)T + (-8.40e3 + 1.45e4i)T^{2} \) |
| 11 | \( 1 + (313. - 542. i)T + (-8.05e4 - 1.39e5i)T^{2} \) |
| 13 | \( 1 + (-117. - 665. i)T + (-3.48e5 + 1.26e5i)T^{2} \) |
| 17 | \( 1 + (-735. + 267. i)T + (1.08e6 - 9.12e5i)T^{2} \) |
| 23 | \( 1 + (-1.88e3 + 1.58e3i)T + (1.11e6 - 6.33e6i)T^{2} \) |
| 29 | \( 1 + (1.42e3 + 517. i)T + (1.57e7 + 1.31e7i)T^{2} \) |
| 31 | \( 1 + (3.99e3 + 6.91e3i)T + (-1.43e7 + 2.47e7i)T^{2} \) |
| 37 | \( 1 + 9.18e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + (-1.15e3 + 6.52e3i)T + (-1.08e8 - 3.96e7i)T^{2} \) |
| 43 | \( 1 + (4.76e3 + 3.99e3i)T + (2.55e7 + 1.44e8i)T^{2} \) |
| 47 | \( 1 + (-3.39e3 - 1.23e3i)T + (1.75e8 + 1.47e8i)T^{2} \) |
| 53 | \( 1 + (-1.59e4 + 1.34e4i)T + (7.26e7 - 4.11e8i)T^{2} \) |
| 59 | \( 1 + (-6.17e3 + 2.24e3i)T + (5.47e8 - 4.59e8i)T^{2} \) |
| 61 | \( 1 + (9.46e3 - 7.93e3i)T + (1.46e8 - 8.31e8i)T^{2} \) |
| 67 | \( 1 + (-5.53e4 - 2.01e4i)T + (1.03e9 + 8.67e8i)T^{2} \) |
| 71 | \( 1 + (9.79e3 + 8.22e3i)T + (3.13e8 + 1.77e9i)T^{2} \) |
| 73 | \( 1 + (7.77e3 - 4.40e4i)T + (-1.94e9 - 7.09e8i)T^{2} \) |
| 79 | \( 1 + (-8.07e3 + 4.57e4i)T + (-2.89e9 - 1.05e9i)T^{2} \) |
| 83 | \( 1 + (2.59e3 + 4.49e3i)T + (-1.96e9 + 3.41e9i)T^{2} \) |
| 89 | \( 1 + (-1.26e4 - 7.19e4i)T + (-5.24e9 + 1.90e9i)T^{2} \) |
| 97 | \( 1 + (-1.18e5 + 4.30e4i)T + (6.57e9 - 5.51e9i)T^{2} \) |
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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
−13.53038279604640259150837970248, −12.73680974817552805068140159504, −11.46630484848299770375872159692, −10.17022011626194830193147089864, −9.026995274176549786851091337903, −7.32167548378150663277988918640, −6.70622162010767738394576617901, −5.37751651881871064619287971482, −2.42030926034061325050657617766, −1.94268677706881359134014158614,
1.15726156728630075387863282632, 3.41955345198634083428054851462, 5.05691532939076201183497805480, 5.66941239546665447153026465996, 8.162433858049168828844726596810, 9.073366315318789586526497555771, 10.23178636119101548797600366072, 10.68572767342413389207930119427, 12.70124933745378864074648569916, 13.60062155149230147045795690008
