| L(s) = 1 | + (6.91 + 5.80i)3-s + (511. − 186. i)5-s + (−283. − 491. i)7-s + (−365. − 2.07e3i)9-s + (2.05e3 − 3.55e3i)11-s + (−5.91e3 + 4.96e3i)13-s + (4.61e3 + 1.68e3i)15-s + (317. − 1.79e3i)17-s + (−1.56e4 + 2.54e4i)19-s + (889. − 5.04e3i)21-s + (−6.25e4 − 2.27e4i)23-s + (1.67e5 − 1.40e5i)25-s + (1.93e4 − 3.35e4i)27-s + (−3.66e4 − 2.08e5i)29-s + (4.05e4 + 7.02e4i)31-s + ⋯ |
| L(s) = 1 | + (0.147 + 0.124i)3-s + (1.83 − 0.666i)5-s + (−0.312 − 0.541i)7-s + (−0.167 − 0.948i)9-s + (0.465 − 0.806i)11-s + (−0.747 + 0.627i)13-s + (0.353 + 0.128i)15-s + (0.0156 − 0.0887i)17-s + (−0.524 + 0.851i)19-s + (0.0209 − 0.118i)21-s + (−1.07 − 0.390i)23-s + (2.14 − 1.79i)25-s + (0.189 − 0.327i)27-s + (−0.279 − 1.58i)29-s + (0.244 + 0.423i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.129 + 0.991i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.129 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.82909 - 1.60557i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.82909 - 1.60557i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (1.56e4 - 2.54e4i)T \) |
| good | 3 | \( 1 + (-6.91 - 5.80i)T + (379. + 2.15e3i)T^{2} \) |
| 5 | \( 1 + (-511. + 186. i)T + (5.98e4 - 5.02e4i)T^{2} \) |
| 7 | \( 1 + (283. + 491. i)T + (-4.11e5 + 7.13e5i)T^{2} \) |
| 11 | \( 1 + (-2.05e3 + 3.55e3i)T + (-9.74e6 - 1.68e7i)T^{2} \) |
| 13 | \( 1 + (5.91e3 - 4.96e3i)T + (1.08e7 - 6.17e7i)T^{2} \) |
| 17 | \( 1 + (-317. + 1.79e3i)T + (-3.85e8 - 1.40e8i)T^{2} \) |
| 23 | \( 1 + (6.25e4 + 2.27e4i)T + (2.60e9 + 2.18e9i)T^{2} \) |
| 29 | \( 1 + (3.66e4 + 2.08e5i)T + (-1.62e10 + 5.89e9i)T^{2} \) |
| 31 | \( 1 + (-4.05e4 - 7.02e4i)T + (-1.37e10 + 2.38e10i)T^{2} \) |
| 37 | \( 1 - 1.40e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + (-9.54e4 - 8.00e4i)T + (3.38e10 + 1.91e11i)T^{2} \) |
| 43 | \( 1 + (-2.53e5 + 9.24e4i)T + (2.08e11 - 1.74e11i)T^{2} \) |
| 47 | \( 1 + (7.99e3 + 4.53e4i)T + (-4.76e11 + 1.73e11i)T^{2} \) |
| 53 | \( 1 + (-4.79e5 - 1.74e5i)T + (8.99e11 + 7.55e11i)T^{2} \) |
| 59 | \( 1 + (-3.36e5 + 1.90e6i)T + (-2.33e12 - 8.51e11i)T^{2} \) |
| 61 | \( 1 + (-1.47e6 - 5.35e5i)T + (2.40e12 + 2.02e12i)T^{2} \) |
| 67 | \( 1 + (-7.59e5 - 4.30e6i)T + (-5.69e12 + 2.07e12i)T^{2} \) |
| 71 | \( 1 + (3.10e6 - 1.13e6i)T + (6.96e12 - 5.84e12i)T^{2} \) |
| 73 | \( 1 + (-1.32e6 - 1.11e6i)T + (1.91e12 + 1.08e13i)T^{2} \) |
| 79 | \( 1 + (-4.13e6 - 3.47e6i)T + (3.33e12 + 1.89e13i)T^{2} \) |
| 83 | \( 1 + (1.23e6 + 2.13e6i)T + (-1.35e13 + 2.35e13i)T^{2} \) |
| 89 | \( 1 + (-2.53e6 + 2.12e6i)T + (7.68e12 - 4.35e13i)T^{2} \) |
| 97 | \( 1 + (4.31e5 - 2.44e6i)T + (-7.59e13 - 2.76e13i)T^{2} \) |
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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
−12.97402100522129468635634826365, −11.88687045007683013246660344246, −10.14736116388743409353621000307, −9.571826994983079120715034590345, −8.546866306138146318947760256059, −6.53765995786141568002979548326, −5.78188937207413642631020285802, −4.10212290252099140186481826257, −2.25568140448162475669318685079, −0.78845929890101509310059470308,
1.89306783750919423869899839392, 2.68616441228525126447697760038, 5.08986207911130051269150356422, 6.14013548479978672673087401521, 7.33339730313913578559670035474, 9.060163921957020005549705756676, 9.942572505435998000270307173886, 10.82408751397590851264422481891, 12.50288922036034892925825892766, 13.38882264128094009166989974080
