| L(s) = 1 | + (7.71 + 13.3i)3-s + (60.2 + 104. i)5-s − 1.39e3·7-s + (974. − 1.68e3i)9-s + 2.04e3·11-s + (4.31e3 − 7.47e3i)13-s + (−929. + 1.60e3i)15-s + (−946. − 1.63e3i)17-s + (1.27e3 + 2.98e4i)19-s + (−1.07e4 − 1.85e4i)21-s + (1.21e4 − 2.11e4i)23-s + (3.17e4 − 5.50e4i)25-s + 6.37e4·27-s + (6.24e4 − 1.08e5i)29-s + 2.12e5·31-s + ⋯ |
| L(s) = 1 | + (0.164 + 0.285i)3-s + (0.215 + 0.373i)5-s − 1.53·7-s + (0.445 − 0.771i)9-s + 0.462·11-s + (0.544 − 0.943i)13-s + (−0.0710 + 0.123i)15-s + (−0.0467 − 0.0809i)17-s + (0.0426 + 0.999i)19-s + (−0.252 − 0.437i)21-s + (0.208 − 0.361i)23-s + (0.407 − 0.705i)25-s + 0.623·27-s + (0.475 − 0.824i)29-s + 1.28·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.768 + 0.639i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.768 + 0.639i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.64322 - 0.594090i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.64322 - 0.594090i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (-1.27e3 - 2.98e4i)T \) |
| good | 3 | \( 1 + (-7.71 - 13.3i)T + (-1.09e3 + 1.89e3i)T^{2} \) |
| 5 | \( 1 + (-60.2 - 104. i)T + (-3.90e4 + 6.76e4i)T^{2} \) |
| 7 | \( 1 + 1.39e3T + 8.23e5T^{2} \) |
| 11 | \( 1 - 2.04e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + (-4.31e3 + 7.47e3i)T + (-3.13e7 - 5.43e7i)T^{2} \) |
| 17 | \( 1 + (946. + 1.63e3i)T + (-2.05e8 + 3.55e8i)T^{2} \) |
| 23 | \( 1 + (-1.21e4 + 2.11e4i)T + (-1.70e9 - 2.94e9i)T^{2} \) |
| 29 | \( 1 + (-6.24e4 + 1.08e5i)T + (-8.62e9 - 1.49e10i)T^{2} \) |
| 31 | \( 1 - 2.12e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 1.17e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + (8.88e4 + 1.53e5i)T + (-9.73e10 + 1.68e11i)T^{2} \) |
| 43 | \( 1 + (6.82e4 + 1.18e5i)T + (-1.35e11 + 2.35e11i)T^{2} \) |
| 47 | \( 1 + (-6.51e5 + 1.12e6i)T + (-2.53e11 - 4.38e11i)T^{2} \) |
| 53 | \( 1 + (-2.33e5 + 4.04e5i)T + (-5.87e11 - 1.01e12i)T^{2} \) |
| 59 | \( 1 + (1.20e6 + 2.08e6i)T + (-1.24e12 + 2.15e12i)T^{2} \) |
| 61 | \( 1 + (5.09e5 - 8.81e5i)T + (-1.57e12 - 2.72e12i)T^{2} \) |
| 67 | \( 1 + (9.92e5 - 1.71e6i)T + (-3.03e12 - 5.24e12i)T^{2} \) |
| 71 | \( 1 + (-2.20e6 - 3.82e6i)T + (-4.54e12 + 7.87e12i)T^{2} \) |
| 73 | \( 1 + (1.07e6 + 1.86e6i)T + (-5.52e12 + 9.56e12i)T^{2} \) |
| 79 | \( 1 + (-4.61e5 - 7.99e5i)T + (-9.60e12 + 1.66e13i)T^{2} \) |
| 83 | \( 1 + 7.74e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + (2.33e6 - 4.03e6i)T + (-2.21e13 - 3.83e13i)T^{2} \) |
| 97 | \( 1 + (8.09e6 + 1.40e7i)T + (-4.03e13 + 6.99e13i)T^{2} \) |
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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
−12.90679278942201291223633038424, −12.05124556640489572732800743487, −10.31824174480254865065550538867, −9.811539942458821803749760813805, −8.527152537314246670860963308310, −6.79635449992865322526919971845, −6.02031751638323908569132635999, −3.94131084739515608792399335325, −2.91141992710475424248530433364, −0.67198368607978002999732885090,
1.23710551923431092077562074918, 2.93595132550684486347880032216, 4.54192436243750120412983200079, 6.25006088943647195224226212436, 7.17855787700879084869110132357, 8.833645920988215117919613218948, 9.640365801921510513863595688768, 10.94555264940653500110312779616, 12.34948970050901521434800917859, 13.25960208533697545405651403552
