| L(s) = 1 | + (10.6 − 11.9i)2-s + (23.9 + 7.77i)3-s + (−27.8 − 254. i)4-s + (−439. − 444. i)5-s + (348. − 201. i)6-s − 3.46e3i·7-s + (−3.32e3 − 2.38e3i)8-s + (−4.79e3 − 3.48e3i)9-s + (−9.98e3 + 484. i)10-s + (1.19e3 + 1.64e3i)11-s + (1.31e3 − 6.30e3i)12-s + (8.83e3 + 6.41e3i)13-s + (−4.12e4 − 3.70e4i)14-s + (−7.05e3 − 1.40e4i)15-s + (−6.39e4 + 1.41e4i)16-s + (3.16e4 + 9.75e4i)17-s + ⋯ |
| L(s) = 1 | + (0.667 − 0.744i)2-s + (0.295 + 0.0959i)3-s + (−0.108 − 0.994i)4-s + (−0.702 − 0.711i)5-s + (0.268 − 0.155i)6-s − 1.44i·7-s + (−0.812 − 0.582i)8-s + (−0.731 − 0.531i)9-s + (−0.998 + 0.0484i)10-s + (0.0814 + 0.112i)11-s + (0.0632 − 0.303i)12-s + (0.309 + 0.224i)13-s + (−1.07 − 0.963i)14-s + (−0.139 − 0.277i)15-s + (−0.976 + 0.216i)16-s + (0.379 + 1.16i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.511 - 0.859i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.511 - 0.859i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.718560 + 1.26369i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.718560 + 1.26369i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-10.6 + 11.9i)T \) |
| 5 | \( 1 + (439. + 444. i)T \) |
| good | 3 | \( 1 + (-23.9 - 7.77i)T + (5.30e3 + 3.85e3i)T^{2} \) |
| 7 | \( 1 + 3.46e3iT - 5.76e6T^{2} \) |
| 11 | \( 1 + (-1.19e3 - 1.64e3i)T + (-6.62e7 + 2.03e8i)T^{2} \) |
| 13 | \( 1 + (-8.83e3 - 6.41e3i)T + (2.52e8 + 7.75e8i)T^{2} \) |
| 17 | \( 1 + (-3.16e4 - 9.75e4i)T + (-5.64e9 + 4.10e9i)T^{2} \) |
| 19 | \( 1 + (-1.62e5 + 5.28e4i)T + (1.37e10 - 9.98e9i)T^{2} \) |
| 23 | \( 1 + (5.68e4 + 7.82e4i)T + (-2.41e10 + 7.44e10i)T^{2} \) |
| 29 | \( 1 + (9.87e4 - 3.03e5i)T + (-4.04e11 - 2.94e11i)T^{2} \) |
| 31 | \( 1 + (8.03e5 - 2.60e5i)T + (6.90e11 - 5.01e11i)T^{2} \) |
| 37 | \( 1 + (-1.01e6 - 7.40e5i)T + (1.08e12 + 3.34e12i)T^{2} \) |
| 41 | \( 1 + (2.18e6 + 1.58e6i)T + (2.46e12 + 7.59e12i)T^{2} \) |
| 43 | \( 1 + 5.79e6iT - 1.16e13T^{2} \) |
| 47 | \( 1 + (3.45e6 + 1.12e6i)T + (1.92e13 + 1.39e13i)T^{2} \) |
| 53 | \( 1 + (1.99e6 - 6.14e6i)T + (-5.03e13 - 3.65e13i)T^{2} \) |
| 59 | \( 1 + (-9.22e6 + 1.26e7i)T + (-4.53e13 - 1.39e14i)T^{2} \) |
| 61 | \( 1 + (-1.63e6 + 1.19e6i)T + (5.92e13 - 1.82e14i)T^{2} \) |
| 67 | \( 1 + (-7.74e6 + 2.51e6i)T + (3.28e14 - 2.38e14i)T^{2} \) |
| 71 | \( 1 + (-6.69e6 - 2.17e6i)T + (5.22e14 + 3.79e14i)T^{2} \) |
| 73 | \( 1 + (1.94e7 - 1.41e7i)T + (2.49e14 - 7.66e14i)T^{2} \) |
| 79 | \( 1 + (-4.62e7 - 1.50e7i)T + (1.22e15 + 8.91e14i)T^{2} \) |
| 83 | \( 1 + (8.40e7 - 2.73e7i)T + (1.82e15 - 1.32e15i)T^{2} \) |
| 89 | \( 1 + (-1.73e7 + 1.25e7i)T + (1.21e15 - 3.74e15i)T^{2} \) |
| 97 | \( 1 + (4.13e6 - 1.27e7i)T + (-6.34e15 - 4.60e15i)T^{2} \) |
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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
−11.63850931166638025018632349874, −10.71711221223950723159481677528, −9.554795579925335826971136099981, −8.392195806266550743449482871195, −6.96643903752090409605216190519, −5.39317988733334262415010570882, −4.03662298827330284072847701888, −3.42809554348290212839517742297, −1.37695450657102910537737164171, −0.32161715382062049410501912800,
2.58439140001314717595830433919, 3.36386176498167440066710327667, 5.14544728321616733628578101077, 6.05501490142481513559993944454, 7.50635556174139228950162340163, 8.242210098349853580483820196852, 9.404787248946518122257549325564, 11.46607559460930210833976702644, 11.79027071217873038233433268488, 13.14487230521993071592435998262
