| L(s) = 1 | + (1.23 − 3.80i)2-s + (−7.28 − 5.29i)3-s + (−12.9 − 9.40i)4-s + (37.7 − 41.2i)5-s + (−29.1 + 21.1i)6-s − 165.·7-s + (−51.7 + 37.6i)8-s + (25.0 + 77.0i)9-s + (−110. − 194. i)10-s + (20.3 − 62.6i)11-s + (44.4 + 136. i)12-s + (−74.3 − 228. i)13-s + (−204. + 628. i)14-s + (−493. + 100. i)15-s + (79.1 + 243. i)16-s + (−1.32e3 + 966. i)17-s + ⋯ |
| L(s) = 1 | + (0.218 − 0.672i)2-s + (−0.467 − 0.339i)3-s + (−0.404 − 0.293i)4-s + (0.675 − 0.737i)5-s + (−0.330 + 0.239i)6-s − 1.27·7-s + (−0.286 + 0.207i)8-s + (0.103 + 0.317i)9-s + (−0.348 − 0.615i)10-s + (0.0507 − 0.156i)11-s + (0.0892 + 0.274i)12-s + (−0.122 − 0.375i)13-s + (−0.278 + 0.857i)14-s + (−0.565 + 0.115i)15-s + (0.0772 + 0.237i)16-s + (−1.11 + 0.810i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.285 - 0.958i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.285 - 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.1585142578\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1585142578\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.23 + 3.80i)T \) |
| 3 | \( 1 + (7.28 + 5.29i)T \) |
| 5 | \( 1 + (-37.7 + 41.2i)T \) |
| good | 7 | \( 1 + 165.T + 1.68e4T^{2} \) |
| 11 | \( 1 + (-20.3 + 62.6i)T + (-1.30e5 - 9.46e4i)T^{2} \) |
| 13 | \( 1 + (74.3 + 228. i)T + (-3.00e5 + 2.18e5i)T^{2} \) |
| 17 | \( 1 + (1.32e3 - 966. i)T + (4.38e5 - 1.35e6i)T^{2} \) |
| 19 | \( 1 + (-436. + 316. i)T + (7.65e5 - 2.35e6i)T^{2} \) |
| 23 | \( 1 + (504. - 1.55e3i)T + (-5.20e6 - 3.78e6i)T^{2} \) |
| 29 | \( 1 + (-1.21e3 - 879. i)T + (6.33e6 + 1.95e7i)T^{2} \) |
| 31 | \( 1 + (-1.31e3 + 954. i)T + (8.84e6 - 2.72e7i)T^{2} \) |
| 37 | \( 1 + (-4.63e3 - 1.42e4i)T + (-5.61e7 + 4.07e7i)T^{2} \) |
| 41 | \( 1 + (-4.30e3 - 1.32e4i)T + (-9.37e7 + 6.80e7i)T^{2} \) |
| 43 | \( 1 + 1.22e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + (-4.53e3 - 3.29e3i)T + (7.08e7 + 2.18e8i)T^{2} \) |
| 53 | \( 1 + (-1.50e4 - 1.09e4i)T + (1.29e8 + 3.97e8i)T^{2} \) |
| 59 | \( 1 + (1.03e4 + 3.18e4i)T + (-5.78e8 + 4.20e8i)T^{2} \) |
| 61 | \( 1 + (5.74e3 - 1.76e4i)T + (-6.83e8 - 4.96e8i)T^{2} \) |
| 67 | \( 1 + (-7.62e3 + 5.53e3i)T + (4.17e8 - 1.28e9i)T^{2} \) |
| 71 | \( 1 + (3.88e4 + 2.82e4i)T + (5.57e8 + 1.71e9i)T^{2} \) |
| 73 | \( 1 + (1.25e4 - 3.86e4i)T + (-1.67e9 - 1.21e9i)T^{2} \) |
| 79 | \( 1 + (4.19e4 + 3.04e4i)T + (9.50e8 + 2.92e9i)T^{2} \) |
| 83 | \( 1 + (8.67e4 - 6.30e4i)T + (1.21e9 - 3.74e9i)T^{2} \) |
| 89 | \( 1 + (-3.58e4 + 1.10e5i)T + (-4.51e9 - 3.28e9i)T^{2} \) |
| 97 | \( 1 + (5.05e4 + 3.67e4i)T + (2.65e9 + 8.16e9i)T^{2} \) |
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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
−12.48742774166263511256080906667, −11.44295591177878421621408242326, −10.21975060648002495207741517274, −9.520464620289510259208306858558, −8.376067191818824528305180018141, −6.59909876331758824022127398791, −5.77893999146694107759421237182, −4.48117201850053779180276125562, −2.87881807694720992962707532288, −1.35434696457794809801713562097,
0.05339763747979590779675211669, 2.63675202750512335086887929029, 4.06562161542681644232862927440, 5.53876785652154789597710873542, 6.51368765140474007175881334710, 7.13787007947481143345680651858, 9.019869759151303993502211654983, 9.763516176495033306915031929368, 10.73100894035348513684284188365, 12.00516580747260176561070168622
