| L(s) = 1 | + (48.0 − 8.47i)3-s + (67.2 − 56.4i)5-s + (−18.9 + 32.8i)7-s + (1.55e3 − 565. i)9-s + (−443. − 767. i)11-s + (2.47e3 + 436. i)13-s + (2.75e3 − 3.28e3i)15-s + (−1.51e3 − 549. i)17-s + (−6.76e3 + 1.10e3i)19-s + (−633. + 1.74e3i)21-s + (1.87e3 + 1.56e3i)23-s + (−1.37e3 + 7.79e3i)25-s + (3.90e4 − 2.25e4i)27-s + (−7.18e3 − 1.97e4i)29-s + (2.33e4 + 1.34e4i)31-s + ⋯ |
| L(s) = 1 | + (1.77 − 0.313i)3-s + (0.538 − 0.451i)5-s + (−0.0553 + 0.0958i)7-s + (2.13 − 0.775i)9-s + (−0.332 − 0.576i)11-s + (1.12 + 0.198i)13-s + (0.816 − 0.972i)15-s + (−0.307 − 0.111i)17-s + (−0.986 + 0.161i)19-s + (−0.0684 + 0.188i)21-s + (0.153 + 0.129i)23-s + (−0.0879 + 0.498i)25-s + (1.98 − 1.14i)27-s + (−0.294 − 0.809i)29-s + (0.782 + 0.451i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.792 + 0.610i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.792 + 0.610i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(3.53550 - 1.20383i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.53550 - 1.20383i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (6.76e3 - 1.10e3i)T \) |
| good | 3 | \( 1 + (-48.0 + 8.47i)T + (685. - 249. i)T^{2} \) |
| 5 | \( 1 + (-67.2 + 56.4i)T + (2.71e3 - 1.53e4i)T^{2} \) |
| 7 | \( 1 + (18.9 - 32.8i)T + (-5.88e4 - 1.01e5i)T^{2} \) |
| 11 | \( 1 + (443. + 767. i)T + (-8.85e5 + 1.53e6i)T^{2} \) |
| 13 | \( 1 + (-2.47e3 - 436. i)T + (4.53e6 + 1.65e6i)T^{2} \) |
| 17 | \( 1 + (1.51e3 + 549. i)T + (1.84e7 + 1.55e7i)T^{2} \) |
| 23 | \( 1 + (-1.87e3 - 1.56e3i)T + (2.57e7 + 1.45e8i)T^{2} \) |
| 29 | \( 1 + (7.18e3 + 1.97e4i)T + (-4.55e8 + 3.82e8i)T^{2} \) |
| 31 | \( 1 + (-2.33e4 - 1.34e4i)T + (4.43e8 + 7.68e8i)T^{2} \) |
| 37 | \( 1 + 2.60e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 + (-4.68e4 + 8.26e3i)T + (4.46e9 - 1.62e9i)T^{2} \) |
| 43 | \( 1 + (-6.62e4 + 5.55e4i)T + (1.09e9 - 6.22e9i)T^{2} \) |
| 47 | \( 1 + (5.40e4 - 1.96e4i)T + (8.25e9 - 6.92e9i)T^{2} \) |
| 53 | \( 1 + (1.61e5 - 1.92e5i)T + (-3.84e9 - 2.18e10i)T^{2} \) |
| 59 | \( 1 + (1.19e5 - 3.26e5i)T + (-3.23e10 - 2.71e10i)T^{2} \) |
| 61 | \( 1 + (2.85e5 + 2.39e5i)T + (8.94e9 + 5.07e10i)T^{2} \) |
| 67 | \( 1 + (-6.20e4 - 1.70e5i)T + (-6.92e10 + 5.81e10i)T^{2} \) |
| 71 | \( 1 + (4.25e5 + 5.06e5i)T + (-2.22e10 + 1.26e11i)T^{2} \) |
| 73 | \( 1 + (-1.15e5 - 6.54e5i)T + (-1.42e11 + 5.17e10i)T^{2} \) |
| 79 | \( 1 + (1.67e5 - 2.94e4i)T + (2.28e11 - 8.31e10i)T^{2} \) |
| 83 | \( 1 + (1.24e5 - 2.15e5i)T + (-1.63e11 - 2.83e11i)T^{2} \) |
| 89 | \( 1 + (-6.92e5 - 1.22e5i)T + (4.67e11 + 1.69e11i)T^{2} \) |
| 97 | \( 1 + (-4.02e4 + 1.10e5i)T + (-6.38e11 - 5.35e11i)T^{2} \) |
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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
−13.47484410710730780263339314575, −12.57916846721156902921456294032, −10.76310184752162116768477134614, −9.315540423353361612798305056894, −8.718565379926903896278763670048, −7.69183078202332374552988467700, −6.13028482621947812078138247144, −4.09728964263256475930366406009, −2.71993838930292771415310435875, −1.40783716597807727894030855260,
1.89566288785879426658625264282, 3.05019520686835423665886541768, 4.37937907208437506743590302108, 6.50057434704495473888224579190, 7.901796408652354249324349773306, 8.831470574229760210752082882322, 9.888236393275996867090134407030, 10.79014278425406522575593587154, 12.84819646935083486631552658308, 13.54983309236782161349780364330
