| L(s) = 1 | + (−1.38 − 7.87i)2-s + (−11.0 + 62.8i)3-s + (−60.1 + 21.8i)4-s + (48.5 + 40.7i)5-s + 510.·6-s + (745. + 625. i)7-s + (256 + 443. i)8-s + (−1.76e3 − 643. i)9-s + (253. − 439. i)10-s + (1.30e3 + 2.26e3i)11-s + (−708. − 4.02e3i)12-s + (1.08e4 − 3.93e3i)13-s + (3.89e3 − 6.74e3i)14-s + (−3.09e3 + 2.60e3i)15-s + (3.13e3 − 2.63e3i)16-s + (−1.09e4 − 3.98e3i)17-s + ⋯ |
| L(s) = 1 | + (−0.122 − 0.696i)2-s + (−0.236 + 1.34i)3-s + (−0.469 + 0.171i)4-s + (0.173 + 0.145i)5-s + 0.964·6-s + (0.821 + 0.689i)7-s + (0.176 + 0.306i)8-s + (−0.808 − 0.294i)9-s + (0.0802 − 0.138i)10-s + (0.296 + 0.513i)11-s + (−0.118 − 0.671i)12-s + (1.36 − 0.496i)13-s + (0.379 − 0.656i)14-s + (−0.237 + 0.198i)15-s + (0.191 − 0.160i)16-s + (−0.540 − 0.196i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.367 - 0.929i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.367 - 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.874461 + 1.28601i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.874461 + 1.28601i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.38 + 7.87i)T \) |
| 37 | \( 1 + (-2.34e5 - 2.00e5i)T \) |
| good | 3 | \( 1 + (11.0 - 62.8i)T + (-2.05e3 - 747. i)T^{2} \) |
| 5 | \( 1 + (-48.5 - 40.7i)T + (1.35e4 + 7.69e4i)T^{2} \) |
| 7 | \( 1 + (-745. - 625. i)T + (1.43e5 + 8.11e5i)T^{2} \) |
| 11 | \( 1 + (-1.30e3 - 2.26e3i)T + (-9.74e6 + 1.68e7i)T^{2} \) |
| 13 | \( 1 + (-1.08e4 + 3.93e3i)T + (4.80e7 - 4.03e7i)T^{2} \) |
| 17 | \( 1 + (1.09e4 + 3.98e3i)T + (3.14e8 + 2.63e8i)T^{2} \) |
| 19 | \( 1 + (3.29e3 - 1.86e4i)T + (-8.39e8 - 3.05e8i)T^{2} \) |
| 23 | \( 1 + (5.15e4 - 8.92e4i)T + (-1.70e9 - 2.94e9i)T^{2} \) |
| 29 | \( 1 + (-4.06e4 - 7.03e4i)T + (-8.62e9 + 1.49e10i)T^{2} \) |
| 31 | \( 1 + 9.34e4T + 2.75e10T^{2} \) |
| 41 | \( 1 + (-7.52e4 + 2.73e4i)T + (1.49e11 - 1.25e11i)T^{2} \) |
| 43 | \( 1 + 5.14e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + (4.43e5 - 7.67e5i)T + (-2.53e11 - 4.38e11i)T^{2} \) |
| 53 | \( 1 + (-1.01e6 + 8.54e5i)T + (2.03e11 - 1.15e12i)T^{2} \) |
| 59 | \( 1 + (6.64e5 - 5.57e5i)T + (4.32e11 - 2.45e12i)T^{2} \) |
| 61 | \( 1 + (3.73e5 - 1.36e5i)T + (2.40e12 - 2.02e12i)T^{2} \) |
| 67 | \( 1 + (3.33e6 + 2.80e6i)T + (1.05e12 + 5.96e12i)T^{2} \) |
| 71 | \( 1 + (-1.29e5 + 7.37e5i)T + (-8.54e12 - 3.11e12i)T^{2} \) |
| 73 | \( 1 + 3.10e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + (-3.49e6 - 2.92e6i)T + (3.33e12 + 1.89e13i)T^{2} \) |
| 83 | \( 1 + (-5.92e6 - 2.15e6i)T + (2.07e13 + 1.74e13i)T^{2} \) |
| 89 | \( 1 + (-3.09e6 + 2.59e6i)T + (7.68e12 - 4.35e13i)T^{2} \) |
| 97 | \( 1 + (-1.06e6 + 1.84e6i)T + (-4.03e13 - 6.99e13i)T^{2} \) |
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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
−13.44954797345657304564337973201, −11.95567438305173741076494294943, −11.12436340086055751288026138962, −10.21856498468618309101008714516, −9.243347121688003734320050832142, −8.151738883114586820081800628650, −5.86604845122200511203459637593, −4.66440888729519306243315689643, −3.51313968292395164558038701522, −1.70442996450660406232049912387,
0.60647509569657964521304559511, 1.71211312012975311157344858111, 4.24753194591520576048315692496, 5.98381443438279603664712926999, 6.82249951928643248015582590071, 7.962487331923821754367812227718, 8.862927846683014247725554007260, 10.74604322131929098938411923247, 11.74727499199929099513482938420, 13.21414585479200670763237543116
