| L(s) = 1 | + (−10.5 − 8.87i)3-s + (−438. + 159. i)5-s + (−408. − 707. i)7-s + (−346. − 1.96e3i)9-s + (−1.15e3 + 2.00e3i)11-s + (2.92e3 − 2.45e3i)13-s + (6.05e3 + 2.20e3i)15-s + (−4.99e3 + 2.83e4i)17-s + (2.90e4 + 7.25e3i)19-s + (−1.95e3 + 1.11e4i)21-s + (−1.83e4 − 6.66e3i)23-s + (1.07e5 − 8.98e4i)25-s + (−2.88e4 + 5.00e4i)27-s + (−2.54e4 − 1.44e5i)29-s + (1.42e5 + 2.47e5i)31-s + ⋯ |
| L(s) = 1 | + (−0.226 − 0.189i)3-s + (−1.56 + 0.571i)5-s + (−0.450 − 0.779i)7-s + (−0.158 − 0.898i)9-s + (−0.262 + 0.454i)11-s + (0.369 − 0.309i)13-s + (0.463 + 0.168i)15-s + (−0.246 + 1.39i)17-s + (0.970 + 0.242i)19-s + (−0.0461 + 0.261i)21-s + (−0.313 − 0.114i)23-s + (1.36 − 1.14i)25-s + (−0.282 + 0.489i)27-s + (−0.193 − 1.09i)29-s + (0.859 + 1.48i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.905 - 0.423i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.905 - 0.423i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.875642 + 0.194544i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.875642 + 0.194544i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (-2.90e4 - 7.25e3i)T \) |
| good | 3 | \( 1 + (10.5 + 8.87i)T + (379. + 2.15e3i)T^{2} \) |
| 5 | \( 1 + (438. - 159. i)T + (5.98e4 - 5.02e4i)T^{2} \) |
| 7 | \( 1 + (408. + 707. i)T + (-4.11e5 + 7.13e5i)T^{2} \) |
| 11 | \( 1 + (1.15e3 - 2.00e3i)T + (-9.74e6 - 1.68e7i)T^{2} \) |
| 13 | \( 1 + (-2.92e3 + 2.45e3i)T + (1.08e7 - 6.17e7i)T^{2} \) |
| 17 | \( 1 + (4.99e3 - 2.83e4i)T + (-3.85e8 - 1.40e8i)T^{2} \) |
| 23 | \( 1 + (1.83e4 + 6.66e3i)T + (2.60e9 + 2.18e9i)T^{2} \) |
| 29 | \( 1 + (2.54e4 + 1.44e5i)T + (-1.62e10 + 5.89e9i)T^{2} \) |
| 31 | \( 1 + (-1.42e5 - 2.47e5i)T + (-1.37e10 + 2.38e10i)T^{2} \) |
| 37 | \( 1 - 2.47e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + (-2.23e5 - 1.87e5i)T + (3.38e10 + 1.91e11i)T^{2} \) |
| 43 | \( 1 + (-2.70e5 + 9.84e4i)T + (2.08e11 - 1.74e11i)T^{2} \) |
| 47 | \( 1 + (-4.45e4 - 2.52e5i)T + (-4.76e11 + 1.73e11i)T^{2} \) |
| 53 | \( 1 + (1.11e6 + 4.04e5i)T + (8.99e11 + 7.55e11i)T^{2} \) |
| 59 | \( 1 + (-2.60e4 + 1.47e5i)T + (-2.33e12 - 8.51e11i)T^{2} \) |
| 61 | \( 1 + (-2.37e6 - 8.64e5i)T + (2.40e12 + 2.02e12i)T^{2} \) |
| 67 | \( 1 + (4.01e5 + 2.27e6i)T + (-5.69e12 + 2.07e12i)T^{2} \) |
| 71 | \( 1 + (5.53e5 - 2.01e5i)T + (6.96e12 - 5.84e12i)T^{2} \) |
| 73 | \( 1 + (-3.10e6 - 2.60e6i)T + (1.91e12 + 1.08e13i)T^{2} \) |
| 79 | \( 1 + (4.73e6 + 3.97e6i)T + (3.33e12 + 1.89e13i)T^{2} \) |
| 83 | \( 1 + (2.68e6 + 4.65e6i)T + (-1.35e13 + 2.35e13i)T^{2} \) |
| 89 | \( 1 + (7.50e6 - 6.29e6i)T + (7.68e12 - 4.35e13i)T^{2} \) |
| 97 | \( 1 + (2.13e6 - 1.20e7i)T + (-7.59e13 - 2.76e13i)T^{2} \) |
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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
−12.91028033168710437281655778830, −12.01238057415705117435738168333, −11.05537336516298832447537151725, −9.992073950946160409979600022529, −8.282610882983479735997413388011, −7.31382507229997332614566723625, −6.26799152106932611122273351930, −4.16611454248517053399166717515, −3.29504544632441740693602825938, −0.76679960050773997308635396290,
0.50734398167225522519000177433, 2.87057231213117224746076713095, 4.39492388251291520341194710545, 5.54735006854537978772783501175, 7.36627295500689328423832720726, 8.348363554823895769794560483577, 9.454480259711185747430804034983, 11.21544545859206956120967197718, 11.67015532032005629455355985198, 12.83808754277297781361845410116
