| L(s) = 1 | + (−3.42 + 5.93i)3-s + (−200. + 346. i)5-s − 1.03e3·7-s + (1.06e3 + 1.85e3i)9-s + 309.·11-s + (−4.89e3 − 8.47e3i)13-s + (−1.37e3 − 2.37e3i)15-s + (9.80e3 − 1.69e4i)17-s + (1.93e4 + 2.27e4i)19-s + (3.54e3 − 6.13e3i)21-s + (−2.63e3 − 4.56e3i)23-s + (−4.09e4 − 7.09e4i)25-s − 2.96e4·27-s + (−9.40e4 − 1.62e5i)29-s − 5.92e4·31-s + ⋯ |
| L(s) = 1 | + (−0.0733 + 0.126i)3-s + (−0.715 + 1.23i)5-s − 1.13·7-s + (0.489 + 0.847i)9-s + 0.0702·11-s + (−0.617 − 1.06i)13-s + (−0.104 − 0.181i)15-s + (0.484 − 0.838i)17-s + (0.648 + 0.761i)19-s + (0.0834 − 0.144i)21-s + (−0.0451 − 0.0782i)23-s + (−0.524 − 0.908i)25-s − 0.290·27-s + (−0.715 − 1.24i)29-s − 0.357·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.128 + 0.991i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.128 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.240115 - 0.273318i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.240115 - 0.273318i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (-1.93e4 - 2.27e4i)T \) |
| good | 3 | \( 1 + (3.42 - 5.93i)T + (-1.09e3 - 1.89e3i)T^{2} \) |
| 5 | \( 1 + (200. - 346. i)T + (-3.90e4 - 6.76e4i)T^{2} \) |
| 7 | \( 1 + 1.03e3T + 8.23e5T^{2} \) |
| 11 | \( 1 - 309.T + 1.94e7T^{2} \) |
| 13 | \( 1 + (4.89e3 + 8.47e3i)T + (-3.13e7 + 5.43e7i)T^{2} \) |
| 17 | \( 1 + (-9.80e3 + 1.69e4i)T + (-2.05e8 - 3.55e8i)T^{2} \) |
| 23 | \( 1 + (2.63e3 + 4.56e3i)T + (-1.70e9 + 2.94e9i)T^{2} \) |
| 29 | \( 1 + (9.40e4 + 1.62e5i)T + (-8.62e9 + 1.49e10i)T^{2} \) |
| 31 | \( 1 + 5.92e4T + 2.75e10T^{2} \) |
| 37 | \( 1 - 6.18e4T + 9.49e10T^{2} \) |
| 41 | \( 1 + (-1.88e5 + 3.25e5i)T + (-9.73e10 - 1.68e11i)T^{2} \) |
| 43 | \( 1 + (-2.76e5 + 4.78e5i)T + (-1.35e11 - 2.35e11i)T^{2} \) |
| 47 | \( 1 + (3.90e5 + 6.76e5i)T + (-2.53e11 + 4.38e11i)T^{2} \) |
| 53 | \( 1 + (5.05e5 + 8.75e5i)T + (-5.87e11 + 1.01e12i)T^{2} \) |
| 59 | \( 1 + (8.84e5 - 1.53e6i)T + (-1.24e12 - 2.15e12i)T^{2} \) |
| 61 | \( 1 + (-7.95e5 - 1.37e6i)T + (-1.57e12 + 2.72e12i)T^{2} \) |
| 67 | \( 1 + (-5.57e5 - 9.65e5i)T + (-3.03e12 + 5.24e12i)T^{2} \) |
| 71 | \( 1 + (1.33e6 - 2.31e6i)T + (-4.54e12 - 7.87e12i)T^{2} \) |
| 73 | \( 1 + (-2.07e6 + 3.58e6i)T + (-5.52e12 - 9.56e12i)T^{2} \) |
| 79 | \( 1 + (1.55e6 - 2.68e6i)T + (-9.60e12 - 1.66e13i)T^{2} \) |
| 83 | \( 1 + 5.31e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + (-4.01e6 - 6.95e6i)T + (-2.21e13 + 3.83e13i)T^{2} \) |
| 97 | \( 1 + (3.68e6 - 6.37e6i)T + (-4.03e13 - 6.99e13i)T^{2} \) |
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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
−12.75045926965883771996527555246, −11.60380826769862176387677097959, −10.41778704378517422041280065411, −9.763247975713166131533787661969, −7.77027942745870875832072355714, −7.09279519601928447625827903265, −5.58177231995265126494084832639, −3.74056829958095609978573637374, −2.66714620194985292635004681015, −0.13831848377551332433832392157,
1.20946303403067686171301937766, 3.49110391615144700923487238337, 4.68486364783514168907870706235, 6.33454889877450933683728578555, 7.53015437520444813295679445792, 9.062019663538425506108611225973, 9.634527100907124051091926970571, 11.44207263161607380396655466821, 12.57663402786100904305155270700, 12.83256837202751028371262572722
