| L(s) = 1 | + (−1.65e10 − 2.87e10i)2-s + (−7.44e17 − 2.31e17i)3-s + (1.83e22 − 3.17e22i)4-s + (−1.05e26 + 1.82e26i)5-s + (5.68e27 + 2.52e28i)6-s + (−1.22e31 − 2.12e31i)7-s + (−2.46e33 − 7.20e16i)8-s + (5.00e35 + 3.45e35i)9-s + 6.97e36·10-s + (−6.94e38 − 1.20e39i)11-s + (−2.10e40 + 1.94e40i)12-s + (−2.26e41 + 3.92e41i)13-s + (−4.07e41 + 7.05e41i)14-s + (1.20e44 − 1.11e44i)15-s + (−6.51e44 − 1.12e45i)16-s + 2.53e46·17-s + ⋯ |
| L(s) = 1 | + (−0.0852 − 0.147i)2-s + (−0.954 − 0.297i)3-s + (0.485 − 0.840i)4-s + (−0.646 + 1.11i)5-s + (0.0375 + 0.166i)6-s + (−0.250 − 0.433i)7-s − 0.336·8-s + (0.823 + 0.567i)9-s + 0.220·10-s + (−0.615 − 1.06i)11-s + (−0.713 + 0.658i)12-s + (−0.382 + 0.662i)13-s + (−0.0426 + 0.0738i)14-s + (0.949 − 0.876i)15-s + (−0.456 − 0.791i)16-s + 1.83·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0944i)\, \overline{\Lambda}(76-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+75/2) \, L(s)\cr =\mathstrut & (0.995 - 0.0944i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(38)\) |
\(\approx\) |
\(1.133598542\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.133598542\) |
| \(L(\frac{77}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (7.44e17 + 2.31e17i)T \) |
| good | 2 | \( 1 + (1.65e10 + 2.87e10i)T + (-1.88e22 + 3.27e22i)T^{2} \) |
| 5 | \( 1 + (1.05e26 - 1.82e26i)T + (-1.32e52 - 2.29e52i)T^{2} \) |
| 7 | \( 1 + (1.22e31 + 2.12e31i)T + (-1.20e63 + 2.08e63i)T^{2} \) |
| 11 | \( 1 + (6.94e38 + 1.20e39i)T + (-6.35e77 + 1.10e78i)T^{2} \) |
| 13 | \( 1 + (2.26e41 - 3.92e41i)T + (-1.75e83 - 3.04e83i)T^{2} \) |
| 17 | \( 1 - 2.53e46T + 1.92e92T^{2} \) |
| 19 | \( 1 - 1.66e48T + 8.06e95T^{2} \) |
| 23 | \( 1 + (3.79e50 - 6.56e50i)T + (-6.73e101 - 1.16e102i)T^{2} \) |
| 29 | \( 1 + (3.08e54 + 5.35e54i)T + (-2.39e109 + 4.14e109i)T^{2} \) |
| 31 | \( 1 + (3.10e55 - 5.37e55i)T + (-3.55e111 - 6.16e111i)T^{2} \) |
| 37 | \( 1 + 7.06e57T + 4.12e117T^{2} \) |
| 41 | \( 1 + (-8.97e59 + 1.55e60i)T + (-4.54e120 - 7.87e120i)T^{2} \) |
| 43 | \( 1 + (-7.89e60 - 1.36e61i)T + (-1.61e122 + 2.80e122i)T^{2} \) |
| 47 | \( 1 + (3.50e60 + 6.06e60i)T + (-1.27e125 + 2.21e125i)T^{2} \) |
| 53 | \( 1 - 2.67e64T + 2.09e129T^{2} \) |
| 59 | \( 1 + (-9.70e65 + 1.68e66i)T + (-3.25e132 - 5.64e132i)T^{2} \) |
| 61 | \( 1 + (-3.90e66 - 6.76e66i)T + (-3.96e133 + 6.87e133i)T^{2} \) |
| 67 | \( 1 + (1.20e68 - 2.09e68i)T + (-4.51e136 - 7.81e136i)T^{2} \) |
| 71 | \( 1 + 3.75e69T + 6.98e138T^{2} \) |
| 73 | \( 1 + 6.75e69T + 5.61e139T^{2} \) |
| 79 | \( 1 + (5.76e70 + 9.98e70i)T + (-1.04e142 + 1.81e142i)T^{2} \) |
| 83 | \( 1 + (-6.68e71 - 1.15e72i)T + (-4.26e143 + 7.38e143i)T^{2} \) |
| 89 | \( 1 + (-6.36e72 + 0.000280i)T + 1.60e146T^{2} \) |
| 97 | \( 1 + (2.75e74 + 4.76e74i)T + (-5.09e148 + 8.81e148i)T^{2} \) |
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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
−10.46824775141785465335284748330, −9.782844907263139871049348771124, −7.56567699508583439265955294781, −7.15895242370624837845033383448, −5.92053508463286857696804178198, −5.32961749059846380490737102130, −3.64671523610971974141024783465, −2.76318559223384143094812876227, −1.35612091570203089309272471817, −0.61298345026792488159040311511,
0.36097251340931682864055888967, 1.35333071634671574349527484119, 2.87566580869294943113241980688, 3.90806866926112576700380702511, 5.04217598283423328169252048421, 5.70020823477437288723089309532, 7.34163328524040443115690330478, 7.80110989934850013268316067387, 9.245449554253672360585489026423, 10.27149390403532680316487906991
