| L(s) = 1 | + (−2 − i)5-s − 2i·7-s + 6·11-s − 2i·13-s + 6i·17-s − 4·19-s + 8i·23-s + (3 + 4i)25-s + 8·31-s + (−2 + 4i)35-s − 2i·37-s + 6·41-s − 4i·43-s − 4i·47-s + 3·49-s + ⋯ |
| L(s) = 1 | + (−0.894 − 0.447i)5-s − 0.755i·7-s + 1.80·11-s − 0.554i·13-s + 1.45i·17-s − 0.917·19-s + 1.66i·23-s + (0.600 + 0.800i)25-s + 1.43·31-s + (−0.338 + 0.676i)35-s − 0.328i·37-s + 0.937·41-s − 0.609i·43-s − 0.583i·47-s + 0.428·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.659983449\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.659983449\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (2 + i)T \) |
| good | 7 | \( 1 + 2iT - 7T^{2} \) |
| 11 | \( 1 - 6T + 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 - 6iT - 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 - 8iT - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 + 4iT - 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 + 6T + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 4T + 71T^{2} \) |
| 73 | \( 1 + 12iT - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 - 12iT - 83T^{2} \) |
| 89 | \( 1 - 14T + 89T^{2} \) |
| 97 | \( 1 - 8iT - 97T^{2} \) |
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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
−8.638286593527112180873910996659, −8.015419910250947460169475604584, −7.25518091034548229923175431757, −6.50746619836270050923751614713, −5.73423023458926962018602847056, −4.52813128498065365848976953081, −3.93346428276504992374475532232, −3.44537823273548394592822463651, −1.73099561547167098771661282987, −0.801120491464059702477752621346,
0.844132592519848614750553438417, 2.34188399939332015494620499759, 3.10626144314218458321137298840, 4.38847534889357960290027505746, 4.48198978094219115269637135009, 6.05091075274299746415282698532, 6.57788988362224368351117355695, 7.17432049229725085986999102289, 8.195119207580441729835707012191, 8.855930421304501323279043789482
