L(s) = 1 | + (39.5 − 68.5i)3-s + (111. − 193. i)5-s − 546.·7-s + (−2.04e3 − 3.53e3i)9-s − 2.77e3·11-s + (−1.14e3 − 1.98e3i)13-s + (−8.82e3 − 1.52e4i)15-s + (1.25e4 − 2.17e4i)17-s + (−1.96e4 + 2.25e4i)19-s + (−2.16e4 + 3.74e4i)21-s + (1.48e4 + 2.56e4i)23-s + (1.42e4 + 2.46e4i)25-s − 1.50e5·27-s + (−4.46e4 − 7.73e4i)29-s + 1.64e5·31-s + ⋯ |
L(s) = 1 | + (0.846 − 1.46i)3-s + (0.398 − 0.690i)5-s − 0.602·7-s + (−0.934 − 1.61i)9-s − 0.628·11-s + (−0.144 − 0.250i)13-s + (−0.675 − 1.17i)15-s + (0.620 − 1.07i)17-s + (−0.655 + 0.755i)19-s + (−0.509 + 0.882i)21-s + (0.253 + 0.439i)23-s + (0.181 + 0.314i)25-s − 1.47·27-s + (−0.340 − 0.589i)29-s + 0.994·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0213i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0213i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(4)\) |
\(\approx\) |
\(0.0200577 - 1.87542i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0200577 - 1.87542i\) |
\(L(\frac{9}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 + (1.96e4 - 2.25e4i)T \) |
good | 3 | \( 1 + (-39.5 + 68.5i)T + (-1.09e3 - 1.89e3i)T^{2} \) |
| 5 | \( 1 + (-111. + 193. i)T + (-3.90e4 - 6.76e4i)T^{2} \) |
| 7 | \( 1 + 546.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 2.77e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + (1.14e3 + 1.98e3i)T + (-3.13e7 + 5.43e7i)T^{2} \) |
| 17 | \( 1 + (-1.25e4 + 2.17e4i)T + (-2.05e8 - 3.55e8i)T^{2} \) |
| 23 | \( 1 + (-1.48e4 - 2.56e4i)T + (-1.70e9 + 2.94e9i)T^{2} \) |
| 29 | \( 1 + (4.46e4 + 7.73e4i)T + (-8.62e9 + 1.49e10i)T^{2} \) |
| 31 | \( 1 - 1.64e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 5.35e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + (4.06e4 - 7.04e4i)T + (-9.73e10 - 1.68e11i)T^{2} \) |
| 43 | \( 1 + (-3.10e5 + 5.37e5i)T + (-1.35e11 - 2.35e11i)T^{2} \) |
| 47 | \( 1 + (3.19e5 + 5.53e5i)T + (-2.53e11 + 4.38e11i)T^{2} \) |
| 53 | \( 1 + (2.27e5 + 3.94e5i)T + (-5.87e11 + 1.01e12i)T^{2} \) |
| 59 | \( 1 + (-3.65e5 + 6.32e5i)T + (-1.24e12 - 2.15e12i)T^{2} \) |
| 61 | \( 1 + (-4.45e5 - 7.71e5i)T + (-1.57e12 + 2.72e12i)T^{2} \) |
| 67 | \( 1 + (-1.77e6 - 3.08e6i)T + (-3.03e12 + 5.24e12i)T^{2} \) |
| 71 | \( 1 + (-2.08e6 + 3.61e6i)T + (-4.54e12 - 7.87e12i)T^{2} \) |
| 73 | \( 1 + (1.65e6 - 2.86e6i)T + (-5.52e12 - 9.56e12i)T^{2} \) |
| 79 | \( 1 + (-1.76e6 + 3.06e6i)T + (-9.60e12 - 1.66e13i)T^{2} \) |
| 83 | \( 1 - 7.47e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + (2.65e6 + 4.59e6i)T + (-2.21e13 + 3.83e13i)T^{2} \) |
| 97 | \( 1 + (-7.83e6 + 1.35e7i)T + (-4.03e13 - 6.99e13i)T^{2} \) |
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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
−12.79675140545806056663392261817, −11.92313268158127439502001658194, −10.00663546096184729491812042283, −8.856598171251019284028530624869, −7.86465557728901786361743043258, −6.80961625364940100169752926751, −5.42955509672084484565877434395, −3.18362940540379997822721406970, −1.89658671742509698087898705090, −0.54197616762041636311137844392,
2.50999834901504682101106491767, 3.51326186877025286553304775848, 4.88262206350997236303824507897, 6.47594608967298040960001927943, 8.194213466202162241269557593983, 9.296403052782176400501214527758, 10.26904481979378221233559395207, 10.81224294111001257331209372285, 12.70502507242768099360344361556, 13.95291711314738221560584958907