| L(s) = 1 | + (7.51 − 2.73i)2-s + (28.9 + 10.5i)3-s + (49.0 − 41.1i)4-s + (−48.7 − 276. i)5-s + 246.·6-s + (−219. − 1.24e3i)7-s + (256. − 443. i)8-s + (−947. − 795. i)9-s + (−1.12e3 − 1.94e3i)10-s + (−3.01e3 + 5.22e3i)11-s + (1.85e3 − 674. i)12-s + (−7.24e3 + 6.07e3i)13-s + (−5.05e3 − 8.75e3i)14-s + (1.50e3 − 8.52e3i)15-s + (711. − 4.03e3i)16-s + (−210. − 176. i)17-s + ⋯ |
| L(s) = 1 | + (0.664 − 0.241i)2-s + (0.619 + 0.225i)3-s + (0.383 − 0.321i)4-s + (−0.174 − 0.989i)5-s + 0.465·6-s + (−0.241 − 1.37i)7-s + (0.176 − 0.306i)8-s + (−0.433 − 0.363i)9-s + (−0.355 − 0.615i)10-s + (−0.683 + 1.18i)11-s + (0.309 − 0.112i)12-s + (−0.914 + 0.766i)13-s + (−0.492 − 0.852i)14-s + (0.114 − 0.651i)15-s + (0.0434 − 0.246i)16-s + (−0.0103 − 0.00870i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.819 + 0.573i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.819 + 0.573i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.643199 - 2.04169i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.643199 - 2.04169i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-7.51 + 2.73i)T \) |
| 37 | \( 1 + (-6.25e4 + 3.01e5i)T \) |
| good | 3 | \( 1 + (-28.9 - 10.5i)T + (1.67e3 + 1.40e3i)T^{2} \) |
| 5 | \( 1 + (48.7 + 276. i)T + (-7.34e4 + 2.67e4i)T^{2} \) |
| 7 | \( 1 + (219. + 1.24e3i)T + (-7.73e5 + 2.81e5i)T^{2} \) |
| 11 | \( 1 + (3.01e3 - 5.22e3i)T + (-9.74e6 - 1.68e7i)T^{2} \) |
| 13 | \( 1 + (7.24e3 - 6.07e3i)T + (1.08e7 - 6.17e7i)T^{2} \) |
| 17 | \( 1 + (210. + 176. i)T + (7.12e7 + 4.04e8i)T^{2} \) |
| 19 | \( 1 + (1.87e4 + 6.82e3i)T + (6.84e8 + 5.74e8i)T^{2} \) |
| 23 | \( 1 + (5.09e3 + 8.82e3i)T + (-1.70e9 + 2.94e9i)T^{2} \) |
| 29 | \( 1 + (-6.21e4 + 1.07e5i)T + (-8.62e9 - 1.49e10i)T^{2} \) |
| 31 | \( 1 - 2.52e5T + 2.75e10T^{2} \) |
| 41 | \( 1 + (-2.71e4 + 2.28e4i)T + (3.38e10 - 1.91e11i)T^{2} \) |
| 43 | \( 1 - 1.91e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + (3.11e5 + 5.40e5i)T + (-2.53e11 + 4.38e11i)T^{2} \) |
| 53 | \( 1 + (-6.97e4 + 3.95e5i)T + (-1.10e12 - 4.01e11i)T^{2} \) |
| 59 | \( 1 + (5.55e4 - 3.15e5i)T + (-2.33e12 - 8.51e11i)T^{2} \) |
| 61 | \( 1 + (-1.57e6 + 1.31e6i)T + (5.45e11 - 3.09e12i)T^{2} \) |
| 67 | \( 1 + (3.54e5 + 2.00e6i)T + (-5.69e12 + 2.07e12i)T^{2} \) |
| 71 | \( 1 + (1.13e6 + 4.13e5i)T + (6.96e12 + 5.84e12i)T^{2} \) |
| 73 | \( 1 - 7.81e4T + 1.10e13T^{2} \) |
| 79 | \( 1 + (-5.40e5 - 3.06e6i)T + (-1.80e13 + 6.56e12i)T^{2} \) |
| 83 | \( 1 + (-6.99e6 - 5.86e6i)T + (4.71e12 + 2.67e13i)T^{2} \) |
| 89 | \( 1 + (-1.44e6 + 8.19e6i)T + (-4.15e13 - 1.51e13i)T^{2} \) |
| 97 | \( 1 + (5.47e6 + 9.47e6i)T + (-4.03e13 + 6.99e13i)T^{2} \) |
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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
−12.78824459021682165343058731033, −11.88222647393360601950381800350, −10.33632846482851448589544533417, −9.444498483197817081348599131030, −7.994696613079853464152768149864, −6.74648601325724013530154458415, −4.78748644158938145720774538116, −4.06824325209370860356476096366, −2.36005804166794176646753151997, −0.50260738162946491454302416117,
2.66556439689707884125834754664, 2.96993060421344279609906898525, 5.26142299351763540120532101247, 6.31469848878554150475007376231, 7.81077694512016852692988918730, 8.655587623139364286921253295235, 10.40766918204209383892463113999, 11.52450671829523774937836840176, 12.65092677452223669752522962480, 13.70440765399936530678362079713
