| L(s) = 1 | + (27.7 + 10.1i)3-s + (5.15 + 29.2i)5-s + (83.7 − 145. i)7-s + (481. + 404. i)9-s + (−29.7 − 51.5i)11-s + (493. − 179. i)13-s + (−152. + 863. i)15-s + (−1.47e3 + 1.23e3i)17-s + (−1.43e3 − 635. i)19-s + (3.79e3 − 3.18e3i)21-s + (−193. + 1.09e3i)23-s + (2.10e3 − 767. i)25-s + (5.70e3 + 9.87e3i)27-s + (−853. − 716. i)29-s + (−2.10e3 + 3.63e3i)31-s + ⋯ |
| L(s) = 1 | + (1.78 + 0.647i)3-s + (0.0922 + 0.523i)5-s + (0.646 − 1.11i)7-s + (1.98 + 1.66i)9-s + (−0.0741 − 0.128i)11-s + (0.810 − 0.294i)13-s + (−0.174 + 0.990i)15-s + (−1.23 + 1.03i)17-s + (−0.914 − 0.403i)19-s + (1.87 − 1.57i)21-s + (−0.0763 + 0.432i)23-s + (0.674 − 0.245i)25-s + (1.50 + 2.60i)27-s + (−0.188 − 0.158i)29-s + (−0.392 + 0.679i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.815 - 0.578i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.815 - 0.578i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.22174 + 1.02552i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.22174 + 1.02552i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (1.43e3 + 635. i)T \) |
| good | 3 | \( 1 + (-27.7 - 10.1i)T + (186. + 156. i)T^{2} \) |
| 5 | \( 1 + (-5.15 - 29.2i)T + (-2.93e3 + 1.06e3i)T^{2} \) |
| 7 | \( 1 + (-83.7 + 145. i)T + (-8.40e3 - 1.45e4i)T^{2} \) |
| 11 | \( 1 + (29.7 + 51.5i)T + (-8.05e4 + 1.39e5i)T^{2} \) |
| 13 | \( 1 + (-493. + 179. i)T + (2.84e5 - 2.38e5i)T^{2} \) |
| 17 | \( 1 + (1.47e3 - 1.23e3i)T + (2.46e5 - 1.39e6i)T^{2} \) |
| 23 | \( 1 + (193. - 1.09e3i)T + (-6.04e6 - 2.20e6i)T^{2} \) |
| 29 | \( 1 + (853. + 716. i)T + (3.56e6 + 2.01e7i)T^{2} \) |
| 31 | \( 1 + (2.10e3 - 3.63e3i)T + (-1.43e7 - 2.47e7i)T^{2} \) |
| 37 | \( 1 - 1.15e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + (1.20e4 + 4.39e3i)T + (8.87e7 + 7.44e7i)T^{2} \) |
| 43 | \( 1 + (3.54e3 + 2.00e4i)T + (-1.38e8 + 5.02e7i)T^{2} \) |
| 47 | \( 1 + (1.77e4 + 1.48e4i)T + (3.98e7 + 2.25e8i)T^{2} \) |
| 53 | \( 1 + (292. - 1.65e3i)T + (-3.92e8 - 1.43e8i)T^{2} \) |
| 59 | \( 1 + (-6.85e3 + 5.75e3i)T + (1.24e8 - 7.04e8i)T^{2} \) |
| 61 | \( 1 + (4.20e3 - 2.38e4i)T + (-7.93e8 - 2.88e8i)T^{2} \) |
| 67 | \( 1 + (4.21e4 + 3.53e4i)T + (2.34e8 + 1.32e9i)T^{2} \) |
| 71 | \( 1 + (1.54e3 + 8.74e3i)T + (-1.69e9 + 6.17e8i)T^{2} \) |
| 73 | \( 1 + (2.04e4 + 7.42e3i)T + (1.58e9 + 1.33e9i)T^{2} \) |
| 79 | \( 1 + (-2.13e4 - 7.76e3i)T + (2.35e9 + 1.97e9i)T^{2} \) |
| 83 | \( 1 + (-2.23e4 + 3.87e4i)T + (-1.96e9 - 3.41e9i)T^{2} \) |
| 89 | \( 1 + (-7.29e4 + 2.65e4i)T + (4.27e9 - 3.58e9i)T^{2} \) |
| 97 | \( 1 + (8.02e4 - 6.73e4i)T + (1.49e9 - 8.45e9i)T^{2} \) |
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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
−13.67728926308131085521562104839, −13.21746386091314573469986828300, −10.83215902110564359432798486450, −10.39731374551486376709096654093, −8.879452866772394717370499865468, −8.134468197884641938195384943492, −6.90018306646826383514282864746, −4.44716932003589049543843792190, −3.48521591333828046547775788350, −1.92040896859847124646930533239,
1.61859400329026231118813331571, 2.71272845896472090969652763669, 4.48936970081457289269990606651, 6.51112784631853625396794374024, 8.015825207936731669732145286104, 8.734177434746464440702192138291, 9.430973584708230155480024777777, 11.42650173261342103743088206849, 12.76287809112591628190541957968, 13.34868466805001007318243677577
