| L(s) = 1 | + (2.30 − 14.5i)2-s + (43.9 − 16.0i)3-s + (−85.1 − 27.6i)4-s + (102. − 260. i)5-s + (−132. − 676. i)6-s + (−331. − 331. i)7-s + (257. − 505. i)8-s + (1.67e3 − 1.40e3i)9-s + (−3.55e3 − 2.08e3i)10-s + (1.79e3 + 2.47e3i)11-s + (−4.18e3 + 150. i)12-s + (796. − 126. i)13-s + (−5.59e3 + 4.06e3i)14-s + (316. − 1.30e4i)15-s + (−1.60e4 − 1.16e4i)16-s + (6.32e3 + 3.22e3i)17-s + ⋯ |
| L(s) = 1 | + (0.203 − 1.28i)2-s + (0.939 − 0.342i)3-s + (−0.665 − 0.216i)4-s + (0.365 − 0.930i)5-s + (−0.250 − 1.27i)6-s + (−0.365 − 0.365i)7-s + (0.177 − 0.348i)8-s + (0.764 − 0.644i)9-s + (−1.12 − 0.660i)10-s + (0.406 + 0.560i)11-s + (−0.699 + 0.0251i)12-s + (0.100 − 0.0159i)13-s + (−0.545 + 0.396i)14-s + (0.0241 − 0.999i)15-s + (−0.978 − 0.711i)16-s + (0.312 + 0.159i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0414i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0414i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.0667576 - 3.22300i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0667576 - 3.22300i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-43.9 + 16.0i)T \) |
| 5 | \( 1 + (-102. + 260. i)T \) |
| good | 2 | \( 1 + (-2.30 + 14.5i)T + (-121. - 39.5i)T^{2} \) |
| 7 | \( 1 + (331. + 331. i)T + 8.23e5iT^{2} \) |
| 11 | \( 1 + (-1.79e3 - 2.47e3i)T + (-6.02e6 + 1.85e7i)T^{2} \) |
| 13 | \( 1 + (-796. + 126. i)T + (5.96e7 - 1.93e7i)T^{2} \) |
| 17 | \( 1 + (-6.32e3 - 3.22e3i)T + (2.41e8 + 3.31e8i)T^{2} \) |
| 19 | \( 1 + (-3.98e3 + 1.29e3i)T + (7.23e8 - 5.25e8i)T^{2} \) |
| 23 | \( 1 + (2.55e4 + 4.04e3i)T + (3.23e9 + 1.05e9i)T^{2} \) |
| 29 | \( 1 + (-3.26e3 + 1.00e4i)T + (-1.39e10 - 1.01e10i)T^{2} \) |
| 31 | \( 1 + (-2.31e4 - 7.11e4i)T + (-2.22e10 + 1.61e10i)T^{2} \) |
| 37 | \( 1 + (-8.35e4 - 5.27e5i)T + (-9.02e10 + 2.93e10i)T^{2} \) |
| 41 | \( 1 + (-4.39e4 + 6.05e4i)T + (-6.01e10 - 1.85e11i)T^{2} \) |
| 43 | \( 1 + (4.62e5 - 4.62e5i)T - 2.71e11iT^{2} \) |
| 47 | \( 1 + (-5.09e4 - 1.00e5i)T + (-2.97e11 + 4.09e11i)T^{2} \) |
| 53 | \( 1 + (1.21e6 - 6.17e5i)T + (6.90e11 - 9.50e11i)T^{2} \) |
| 59 | \( 1 + (-1.86e6 - 1.35e6i)T + (7.69e11 + 2.36e12i)T^{2} \) |
| 61 | \( 1 + (-2.57e6 + 1.87e6i)T + (9.71e11 - 2.98e12i)T^{2} \) |
| 67 | \( 1 + (-1.64e6 + 3.23e6i)T + (-3.56e12 - 4.90e12i)T^{2} \) |
| 71 | \( 1 + (-2.43e6 - 7.90e5i)T + (7.35e12 + 5.34e12i)T^{2} \) |
| 73 | \( 1 + (-1.97e5 + 1.24e6i)T + (-1.05e13 - 3.41e12i)T^{2} \) |
| 79 | \( 1 + (1.48e6 + 4.81e5i)T + (1.55e13 + 1.12e13i)T^{2} \) |
| 83 | \( 1 + (2.93e6 - 5.76e6i)T + (-1.59e13 - 2.19e13i)T^{2} \) |
| 89 | \( 1 + (-7.85e6 + 5.70e6i)T + (1.36e13 - 4.20e13i)T^{2} \) |
| 97 | \( 1 + (-3.07e6 + 1.56e6i)T + (4.74e13 - 6.53e13i)T^{2} \) |
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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
−12.64628410147611954835224468552, −11.75968785129595480187422722264, −10.08418696380701423800297849358, −9.508644717223807314354392817784, −8.201511827462127577565565678974, −6.70276281968301363623402382886, −4.55492927911960197149130353442, −3.38353917741737312549853776624, −1.95593227833125587885491438718, −0.955435981402420223290017580396,
2.28423722939962495700806136935, 3.72495697573759337826230410135, 5.52469425051367488955088948182, 6.65109824183115511318728321467, 7.69806142397336925937275570940, 8.846052837805816577546599477705, 9.993280117219403490561703508444, 11.28910917202113575856350069202, 13.15183410847107705878733723961, 14.18177678890632719239531632596
