| L(s) = 1 | + (3.99 + 3.35i)3-s + (24.8 − 9.05i)5-s + (24.2 + 41.9i)7-s + (−375. − 2.12e3i)9-s + (−3.46e3 + 6.00e3i)11-s + (4.02e3 − 3.37e3i)13-s + (129. + 47.2i)15-s + (2.34e3 − 1.32e4i)17-s + (−2.95e4 − 4.33e3i)19-s + (−43.8 + 248. i)21-s + (−5.91e3 − 2.15e3i)23-s + (−5.93e4 + 4.97e4i)25-s + (1.13e4 − 1.96e4i)27-s + (−2.18e4 − 1.24e5i)29-s + (−1.13e5 − 1.96e5i)31-s + ⋯ |
| L(s) = 1 | + (0.0854 + 0.0716i)3-s + (0.0889 − 0.0323i)5-s + (0.0266 + 0.0462i)7-s + (−0.171 − 0.972i)9-s + (−0.784 + 1.35i)11-s + (0.508 − 0.426i)13-s + (0.00992 + 0.00361i)15-s + (0.115 − 0.656i)17-s + (−0.989 − 0.144i)19-s + (−0.00103 + 0.00586i)21-s + (−0.101 − 0.0368i)23-s + (−0.759 + 0.637i)25-s + (0.110 − 0.191i)27-s + (−0.166 − 0.944i)29-s + (−0.685 − 1.18i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.859 + 0.511i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.859 + 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.141215 - 0.513379i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.141215 - 0.513379i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (2.95e4 + 4.33e3i)T \) |
| good | 3 | \( 1 + (-3.99 - 3.35i)T + (379. + 2.15e3i)T^{2} \) |
| 5 | \( 1 + (-24.8 + 9.05i)T + (5.98e4 - 5.02e4i)T^{2} \) |
| 7 | \( 1 + (-24.2 - 41.9i)T + (-4.11e5 + 7.13e5i)T^{2} \) |
| 11 | \( 1 + (3.46e3 - 6.00e3i)T + (-9.74e6 - 1.68e7i)T^{2} \) |
| 13 | \( 1 + (-4.02e3 + 3.37e3i)T + (1.08e7 - 6.17e7i)T^{2} \) |
| 17 | \( 1 + (-2.34e3 + 1.32e4i)T + (-3.85e8 - 1.40e8i)T^{2} \) |
| 23 | \( 1 + (5.91e3 + 2.15e3i)T + (2.60e9 + 2.18e9i)T^{2} \) |
| 29 | \( 1 + (2.18e4 + 1.24e5i)T + (-1.62e10 + 5.89e9i)T^{2} \) |
| 31 | \( 1 + (1.13e5 + 1.96e5i)T + (-1.37e10 + 2.38e10i)T^{2} \) |
| 37 | \( 1 + 4.57e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + (7.33e4 + 6.15e4i)T + (3.38e10 + 1.91e11i)T^{2} \) |
| 43 | \( 1 + (6.37e5 - 2.32e5i)T + (2.08e11 - 1.74e11i)T^{2} \) |
| 47 | \( 1 + (1.41e5 + 8.00e5i)T + (-4.76e11 + 1.73e11i)T^{2} \) |
| 53 | \( 1 + (1.38e6 + 5.04e5i)T + (8.99e11 + 7.55e11i)T^{2} \) |
| 59 | \( 1 + (4.98e5 - 2.82e6i)T + (-2.33e12 - 8.51e11i)T^{2} \) |
| 61 | \( 1 + (-1.54e6 - 5.63e5i)T + (2.40e12 + 2.02e12i)T^{2} \) |
| 67 | \( 1 + (6.70e5 + 3.80e6i)T + (-5.69e12 + 2.07e12i)T^{2} \) |
| 71 | \( 1 + (-2.35e6 + 8.57e5i)T + (6.96e12 - 5.84e12i)T^{2} \) |
| 73 | \( 1 + (-3.47e6 - 2.91e6i)T + (1.91e12 + 1.08e13i)T^{2} \) |
| 79 | \( 1 + (-5.84e6 - 4.90e6i)T + (3.33e12 + 1.89e13i)T^{2} \) |
| 83 | \( 1 + (-2.96e6 - 5.13e6i)T + (-1.35e13 + 2.35e13i)T^{2} \) |
| 89 | \( 1 + (-3.99e6 + 3.35e6i)T + (7.68e12 - 4.35e13i)T^{2} \) |
| 97 | \( 1 + (-4.22e5 + 2.39e6i)T + (-7.59e13 - 2.76e13i)T^{2} \) |
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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
−12.64200392183521262738143099937, −11.60380845870932447922535363814, −10.23674549259220348111529037533, −9.317097062085918268164115818046, −7.973963445585972827925663622869, −6.69590628829058716140449941989, −5.28221596806742447714039369724, −3.77087617130207907866664475394, −2.11433169024748893305294246085, −0.16621763165886009324057100711,
1.82129550046870592375115628360, 3.42219952256405776228366079512, 5.13721421255317877637226013811, 6.35941359552445004834531434033, 7.979294758045320586402355079719, 8.737793791362858966723531226593, 10.49369332071913246001624894037, 11.04669988502883306199372059521, 12.57437577884695813419748049936, 13.61347635106241367867365419889
