| L(s) = 1 | + (45.2 + 78.3i)2-s + (−3.14e3 − 1.81e3i)3-s + (−4.09e3 + 7.09e3i)4-s + (−1.12e5 + 6.49e4i)5-s − 3.29e5i·6-s + (−6.29e5 + 5.30e5i)7-s − 7.41e5·8-s + (4.22e6 + 7.30e6i)9-s + (−1.01e7 − 5.87e6i)10-s + (8.48e6 − 1.46e7i)11-s + (2.57e7 − 1.48e7i)12-s − 7.99e7i·13-s + (−7.01e7 − 2.53e7i)14-s + 4.72e8·15-s + (−3.35e7 − 5.81e7i)16-s + (1.21e8 + 7.04e7i)17-s + ⋯ |
| L(s) = 1 | + (0.353 + 0.612i)2-s + (−1.43 − 0.831i)3-s + (−0.249 + 0.433i)4-s + (−1.43 + 0.831i)5-s − 1.17i·6-s + (−0.764 + 0.644i)7-s − 0.353·8-s + (0.882 + 1.52i)9-s + (−1.01 − 0.587i)10-s + (0.435 − 0.753i)11-s + (0.719 − 0.415i)12-s − 1.27i·13-s + (−0.665 − 0.240i)14-s + 2.76·15-s + (−0.125 − 0.216i)16-s + (0.297 + 0.171i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.975 + 0.217i)\, \overline{\Lambda}(15-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+7) \, L(s)\cr =\mathstrut & (0.975 + 0.217i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{15}{2})\) |
\(\approx\) |
\(0.458079 - 0.0505342i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.458079 - 0.0505342i\) |
| \(L(8)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-45.2 - 78.3i)T \) |
| 7 | \( 1 + (6.29e5 - 5.30e5i)T \) |
| good | 3 | \( 1 + (3.14e3 + 1.81e3i)T + (2.39e6 + 4.14e6i)T^{2} \) |
| 5 | \( 1 + (1.12e5 - 6.49e4i)T + (3.05e9 - 5.28e9i)T^{2} \) |
| 11 | \( 1 + (-8.48e6 + 1.46e7i)T + (-1.89e14 - 3.28e14i)T^{2} \) |
| 13 | \( 1 + 7.99e7iT - 3.93e15T^{2} \) |
| 17 | \( 1 + (-1.21e8 - 7.04e7i)T + (8.41e16 + 1.45e17i)T^{2} \) |
| 19 | \( 1 + (1.12e9 - 6.51e8i)T + (3.99e17 - 6.91e17i)T^{2} \) |
| 23 | \( 1 + (-2.23e9 - 3.87e9i)T + (-5.79e18 + 1.00e19i)T^{2} \) |
| 29 | \( 1 + 1.38e10T + 2.97e20T^{2} \) |
| 31 | \( 1 + (-1.21e10 - 6.99e9i)T + (3.78e20 + 6.55e20i)T^{2} \) |
| 37 | \( 1 + (-1.37e10 - 2.38e10i)T + (-4.50e21 + 7.80e21i)T^{2} \) |
| 41 | \( 1 + 1.39e11iT - 3.79e22T^{2} \) |
| 43 | \( 1 - 1.05e11T + 7.38e22T^{2} \) |
| 47 | \( 1 + (2.06e11 - 1.19e11i)T + (1.28e23 - 2.22e23i)T^{2} \) |
| 53 | \( 1 + (-3.56e11 + 6.18e11i)T + (-6.89e23 - 1.19e24i)T^{2} \) |
| 59 | \( 1 + (2.66e12 + 1.53e12i)T + (3.09e24 + 5.36e24i)T^{2} \) |
| 61 | \( 1 + (-2.59e12 + 1.49e12i)T + (4.93e24 - 8.55e24i)T^{2} \) |
| 67 | \( 1 + (2.00e12 - 3.47e12i)T + (-1.83e25 - 3.18e25i)T^{2} \) |
| 71 | \( 1 - 1.48e13T + 8.27e25T^{2} \) |
| 73 | \( 1 + (3.42e12 + 1.97e12i)T + (6.10e25 + 1.05e26i)T^{2} \) |
| 79 | \( 1 + (3.03e12 + 5.26e12i)T + (-1.84e26 + 3.19e26i)T^{2} \) |
| 83 | \( 1 - 9.40e12iT - 7.36e26T^{2} \) |
| 89 | \( 1 + (-6.63e13 + 3.83e13i)T + (9.78e26 - 1.69e27i)T^{2} \) |
| 97 | \( 1 - 8.19e13iT - 6.52e27T^{2} \) |
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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
−16.05308857117484715166232567962, −15.03061565612425457570791868301, −12.92069749184841795380226199735, −11.98408085152147174888497167571, −10.88001884472765781727300762668, −7.976804369341430569908377283254, −6.71360384238425665236446048037, −5.66798075733678495694583710366, −3.45820286987485807965974003292, −0.34456148547312468466968508363,
0.66757527708247795920912827642, 4.09692721596603642503011994000, 4.59305356106094340775966588735, 6.70325720255965892038867329153, 9.304342694650313931349932482693, 10.82132362643781939256712735590, 11.82473425997626343049461051629, 12.73501325477048977749065806609, 15.11899621482472135860606161589, 16.35995332426939417573390064791
