| L(s) = 1 | + (−16.9 + 12.3i)2-s + (96.2 − 296. i)4-s + (380. + 276. i)5-s + (39.7 − 122. i)7-s + (1.18e3 + 3.66e3i)8-s − 9.86e3·10-s + (−4.28e3 + 1.07e3i)11-s + (7.49e3 − 5.44e3i)13-s + (834. + 2.56e3i)14-s + (−3.30e4 − 2.40e4i)16-s + (−6.11e3 − 4.43e3i)17-s + (1.30e4 + 4.00e4i)19-s + (1.18e5 − 8.61e4i)20-s + (5.94e4 − 7.09e4i)22-s + 5.83e4·23-s + ⋯ |
| L(s) = 1 | + (−1.49 + 1.08i)2-s + (0.752 − 2.31i)4-s + (1.36 + 0.989i)5-s + (0.0438 − 0.134i)7-s + (0.821 + 2.52i)8-s − 3.11·10-s + (−0.969 + 0.243i)11-s + (0.946 − 0.687i)13-s + (0.0812 + 0.250i)14-s + (−2.01 − 1.46i)16-s + (−0.301 − 0.219i)17-s + (0.435 + 1.33i)19-s + (3.31 − 2.40i)20-s + (1.18 − 1.42i)22-s + 1.00·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.510 - 0.859i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.510 - 0.859i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.580032 + 1.01914i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.580032 + 1.01914i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 + (4.28e3 - 1.07e3i)T \) |
| good | 2 | \( 1 + (16.9 - 12.3i)T + (39.5 - 121. i)T^{2} \) |
| 5 | \( 1 + (-380. - 276. i)T + (2.41e4 + 7.43e4i)T^{2} \) |
| 7 | \( 1 + (-39.7 + 122. i)T + (-6.66e5 - 4.84e5i)T^{2} \) |
| 13 | \( 1 + (-7.49e3 + 5.44e3i)T + (1.93e7 - 5.96e7i)T^{2} \) |
| 17 | \( 1 + (6.11e3 + 4.43e3i)T + (1.26e8 + 3.90e8i)T^{2} \) |
| 19 | \( 1 + (-1.30e4 - 4.00e4i)T + (-7.23e8 + 5.25e8i)T^{2} \) |
| 23 | \( 1 - 5.83e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + (-1.61e4 + 4.95e4i)T + (-1.39e10 - 1.01e10i)T^{2} \) |
| 31 | \( 1 + (-2.52e5 + 1.83e5i)T + (8.50e9 - 2.61e10i)T^{2} \) |
| 37 | \( 1 + (-4.16e4 + 1.28e5i)T + (-7.68e10 - 5.57e10i)T^{2} \) |
| 41 | \( 1 + (4.27e4 + 1.31e5i)T + (-1.57e11 + 1.14e11i)T^{2} \) |
| 43 | \( 1 + 5.43e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + (-2.57e5 - 7.91e5i)T + (-4.09e11 + 2.97e11i)T^{2} \) |
| 53 | \( 1 + (4.44e5 - 3.23e5i)T + (3.63e11 - 1.11e12i)T^{2} \) |
| 59 | \( 1 + (-1.20e5 + 3.70e5i)T + (-2.01e12 - 1.46e12i)T^{2} \) |
| 61 | \( 1 + (-2.10e6 - 1.53e6i)T + (9.71e11 + 2.98e12i)T^{2} \) |
| 67 | \( 1 + 7.79e5T + 6.06e12T^{2} \) |
| 71 | \( 1 + (-1.68e6 - 1.22e6i)T + (2.81e12 + 8.64e12i)T^{2} \) |
| 73 | \( 1 + (1.00e6 - 3.09e6i)T + (-8.93e12 - 6.49e12i)T^{2} \) |
| 79 | \( 1 + (5.55e6 - 4.03e6i)T + (5.93e12 - 1.82e13i)T^{2} \) |
| 83 | \( 1 + (-6.37e5 - 4.63e5i)T + (8.38e12 + 2.58e13i)T^{2} \) |
| 89 | \( 1 + 3.39e5T + 4.42e13T^{2} \) |
| 97 | \( 1 + (-5.64e6 + 4.10e6i)T + (2.49e13 - 7.68e13i)T^{2} \) |
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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
−13.27642465757102208103252228822, −11.04670876545092569528554788060, −10.25762265076817805709830598841, −9.696811097183156795179528337341, −8.365939605181889167077281018929, −7.34976582753462815995820850457, −6.23507094803725398203684469032, −5.54337157250921942826363801521, −2.54396889296797507172581321618, −1.11792574907135750809748287581,
0.71710392324508630371026083726, 1.71283477288139151178912534030, 2.89037684776440904885606523651, 5.00011492571235558892259714072, 6.72538526110619758770539730981, 8.470799038415126073053843820112, 8.916792847491077681496969780973, 9.913682384748145364725014610134, 10.79760516613962799752171440450, 11.86034577568576035046822804639
