| L(s) = 1 | + (−22.8 + 8.31i)3-s + (−73.1 + 415. i)5-s + (−566. − 980. i)7-s + (−1.22e3 + 1.02e3i)9-s + (−652. + 1.12e3i)11-s + (−4.78e3 − 1.74e3i)13-s + (−1.77e3 − 1.00e4i)15-s + (2.05e4 + 1.72e4i)17-s + (2.89e4 + 7.55e3i)19-s + (2.10e4 + 1.76e4i)21-s + (−1.83e4 − 1.03e5i)23-s + (−9.34e4 − 3.40e4i)25-s + (4.59e4 − 7.96e4i)27-s + (−9.61e3 + 8.06e3i)29-s + (−1.20e5 − 2.08e5i)31-s + ⋯ |
| L(s) = 1 | + (−0.488 + 0.177i)3-s + (−0.261 + 1.48i)5-s + (−0.623 − 1.08i)7-s + (−0.558 + 0.468i)9-s + (−0.147 + 0.255i)11-s + (−0.604 − 0.219i)13-s + (−0.136 − 0.772i)15-s + (1.01 + 0.851i)17-s + (0.967 + 0.252i)19-s + (0.496 + 0.416i)21-s + (−0.313 − 1.78i)23-s + (−1.19 − 0.435i)25-s + (0.449 − 0.778i)27-s + (−0.0731 + 0.0614i)29-s + (−0.725 − 1.25i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.279 + 0.960i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.279 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.458678 - 0.344250i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.458678 - 0.344250i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (-2.89e4 - 7.55e3i)T \) |
| good | 3 | \( 1 + (22.8 - 8.31i)T + (1.67e3 - 1.40e3i)T^{2} \) |
| 5 | \( 1 + (73.1 - 415. i)T + (-7.34e4 - 2.67e4i)T^{2} \) |
| 7 | \( 1 + (566. + 980. i)T + (-4.11e5 + 7.13e5i)T^{2} \) |
| 11 | \( 1 + (652. - 1.12e3i)T + (-9.74e6 - 1.68e7i)T^{2} \) |
| 13 | \( 1 + (4.78e3 + 1.74e3i)T + (4.80e7 + 4.03e7i)T^{2} \) |
| 17 | \( 1 + (-2.05e4 - 1.72e4i)T + (7.12e7 + 4.04e8i)T^{2} \) |
| 23 | \( 1 + (1.83e4 + 1.03e5i)T + (-3.19e9 + 1.16e9i)T^{2} \) |
| 29 | \( 1 + (9.61e3 - 8.06e3i)T + (2.99e9 - 1.69e10i)T^{2} \) |
| 31 | \( 1 + (1.20e5 + 2.08e5i)T + (-1.37e10 + 2.38e10i)T^{2} \) |
| 37 | \( 1 + 5.53e4T + 9.49e10T^{2} \) |
| 41 | \( 1 + (-6.82e5 + 2.48e5i)T + (1.49e11 - 1.25e11i)T^{2} \) |
| 43 | \( 1 + (-2.47e4 + 1.40e5i)T + (-2.55e11 - 9.29e10i)T^{2} \) |
| 47 | \( 1 + (2.79e5 - 2.34e5i)T + (8.79e10 - 4.98e11i)T^{2} \) |
| 53 | \( 1 + (3.43e5 + 1.94e6i)T + (-1.10e12 + 4.01e11i)T^{2} \) |
| 59 | \( 1 + (-1.72e6 - 1.44e6i)T + (4.32e11 + 2.45e12i)T^{2} \) |
| 61 | \( 1 + (-6.85e4 - 3.88e5i)T + (-2.95e12 + 1.07e12i)T^{2} \) |
| 67 | \( 1 + (-1.35e4 + 1.13e4i)T + (1.05e12 - 5.96e12i)T^{2} \) |
| 71 | \( 1 + (7.63e4 - 4.33e5i)T + (-8.54e12 - 3.11e12i)T^{2} \) |
| 73 | \( 1 + (4.94e6 - 1.79e6i)T + (8.46e12 - 7.10e12i)T^{2} \) |
| 79 | \( 1 + (3.46e6 - 1.26e6i)T + (1.47e13 - 1.23e13i)T^{2} \) |
| 83 | \( 1 + (-2.07e6 - 3.59e6i)T + (-1.35e13 + 2.35e13i)T^{2} \) |
| 89 | \( 1 + (8.17e6 + 2.97e6i)T + (3.38e13 + 2.84e13i)T^{2} \) |
| 97 | \( 1 + (-2.03e6 - 1.70e6i)T + (1.40e13 + 7.95e13i)T^{2} \) |
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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
−12.80899744111795875382840023663, −11.50389298431576117646569949117, −10.48971879388420952495471398752, −10.04382077074806552002223357739, −7.87321499934135544348913752234, −6.97293229074105545913547296150, −5.76361413544924558930890878678, −3.95433386818457262554558755264, −2.68741081514079170688058284540, −0.24376934363852609168114127285,
1.08355679682283643880563522914, 3.15799584767255863651101388020, 5.13503708372922681057225315070, 5.77456245595087288260152256543, 7.55761040538015383376631687591, 8.993054222536941634071702227819, 9.523391818233468458775124562776, 11.62761686544400192151097486117, 12.10388099363181598926833269792, 12.94787183018800326776048328830
