| L(s) = 1 | + (13.6 − 7.88i)3-s + (93.3 + 161. i)5-s − 434.·7-s + (−240. + 415. i)9-s − 255.·11-s + (−2.46e3 − 1.42e3i)13-s + (2.54e3 + 1.47e3i)15-s + (−3.36e3 − 5.82e3i)17-s + (3.74e3 + 5.74e3i)19-s + (−5.93e3 + 3.42e3i)21-s + (−5.56e3 + 9.64e3i)23-s + (−9.60e3 + 1.66e4i)25-s + 1.90e4i·27-s + (−1.65e4 − 9.55e3i)29-s − 6.02e3i·31-s + ⋯ |
| L(s) = 1 | + (0.505 − 0.292i)3-s + (0.746 + 1.29i)5-s − 1.26·7-s + (−0.329 + 0.570i)9-s − 0.191·11-s + (−1.12 − 0.649i)13-s + (0.755 + 0.436i)15-s + (−0.684 − 1.18i)17-s + (0.546 + 0.837i)19-s + (−0.641 + 0.370i)21-s + (−0.457 + 0.792i)23-s + (−0.614 + 1.06i)25-s + 0.969i·27-s + (−0.678 − 0.391i)29-s − 0.202i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.843 - 0.536i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.843 - 0.536i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.259472 + 0.890994i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.259472 + 0.890994i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (-3.74e3 - 5.74e3i)T \) |
| good | 3 | \( 1 + (-13.6 + 7.88i)T + (364.5 - 631. i)T^{2} \) |
| 5 | \( 1 + (-93.3 - 161. i)T + (-7.81e3 + 1.35e4i)T^{2} \) |
| 7 | \( 1 + 434.T + 1.17e5T^{2} \) |
| 11 | \( 1 + 255.T + 1.77e6T^{2} \) |
| 13 | \( 1 + (2.46e3 + 1.42e3i)T + (2.41e6 + 4.18e6i)T^{2} \) |
| 17 | \( 1 + (3.36e3 + 5.82e3i)T + (-1.20e7 + 2.09e7i)T^{2} \) |
| 23 | \( 1 + (5.56e3 - 9.64e3i)T + (-7.40e7 - 1.28e8i)T^{2} \) |
| 29 | \( 1 + (1.65e4 + 9.55e3i)T + (2.97e8 + 5.15e8i)T^{2} \) |
| 31 | \( 1 + 6.02e3iT - 8.87e8T^{2} \) |
| 37 | \( 1 - 3.62e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 + (7.66e4 - 4.42e4i)T + (2.37e9 - 4.11e9i)T^{2} \) |
| 43 | \( 1 + (-7.10e4 - 1.23e5i)T + (-3.16e9 + 5.47e9i)T^{2} \) |
| 47 | \( 1 + (-1.72e4 + 2.97e4i)T + (-5.38e9 - 9.33e9i)T^{2} \) |
| 53 | \( 1 + (-1.73e5 - 1.00e5i)T + (1.10e10 + 1.91e10i)T^{2} \) |
| 59 | \( 1 + (-9.36e4 + 5.40e4i)T + (2.10e10 - 3.65e10i)T^{2} \) |
| 61 | \( 1 + (-1.76e5 + 3.06e5i)T + (-2.57e10 - 4.46e10i)T^{2} \) |
| 67 | \( 1 + (-2.97e5 - 1.71e5i)T + (4.52e10 + 7.83e10i)T^{2} \) |
| 71 | \( 1 + (2.72e5 - 1.57e5i)T + (6.40e10 - 1.10e11i)T^{2} \) |
| 73 | \( 1 + (-1.16e4 - 2.01e4i)T + (-7.56e10 + 1.31e11i)T^{2} \) |
| 79 | \( 1 + (1.10e5 - 6.40e4i)T + (1.21e11 - 2.10e11i)T^{2} \) |
| 83 | \( 1 - 4.33e5T + 3.26e11T^{2} \) |
| 89 | \( 1 + (4.90e5 + 2.83e5i)T + (2.48e11 + 4.30e11i)T^{2} \) |
| 97 | \( 1 + (1.34e6 - 7.78e5i)T + (4.16e11 - 7.21e11i)T^{2} \) |
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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
−13.74816007560413388413715709795, −12.98963377047474826866572971333, −11.47598760943445369371657150113, −10.12348460616217935742895512527, −9.568068102951313107264162467351, −7.73957045368302220343279764216, −6.79727821261409252603866390043, −5.53548812381020342958120060818, −3.13818357294520126048800797786, −2.38912845659969279232769124854,
0.29728475631547845227186954640, 2.30931977752403383909887841011, 3.99711385446045017261962882608, 5.49273101725819438307532842128, 6.80039293124696725905905581190, 8.740539187164036786961679032111, 9.263265990974953904437189525922, 10.20448719233809274435907123316, 12.12176065825567848470936193096, 12.87197434297418685101349971826
