| L(s) = 1 | + (19.7 − 60.8i)2-s + (1.76e3 − 1.28e3i)3-s + (−3.31e3 − 2.40e3i)4-s + (6.22e3 + 1.91e4i)5-s + (−4.31e4 − 1.32e5i)6-s + (3.39e5 + 2.46e5i)7-s + (−2.12e5 + 1.54e5i)8-s + (9.80e5 − 3.01e6i)9-s + 1.28e6·10-s + (5.36e6 − 2.38e6i)11-s − 8.94e6·12-s + (1.01e7 − 3.11e7i)13-s + (2.17e7 − 1.58e7i)14-s + (3.55e7 + 2.58e7i)15-s + (5.18e6 + 1.59e7i)16-s + (1.32e7 + 4.06e7i)17-s + ⋯ |
| L(s) = 1 | + (0.218 − 0.672i)2-s + (1.39 − 1.01i)3-s + (−0.404 − 0.293i)4-s + (0.178 + 0.548i)5-s + (−0.377 − 1.16i)6-s + (1.09 + 0.793i)7-s + (−0.286 + 0.207i)8-s + (0.614 − 1.89i)9-s + 0.407·10-s + (0.913 − 0.406i)11-s − 0.864·12-s + (0.581 − 1.78i)13-s + (0.772 − 0.560i)14-s + (0.806 + 0.585i)15-s + (0.0772 + 0.237i)16-s + (0.132 + 0.408i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.163 + 0.986i)\, \overline{\Lambda}(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (-0.163 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(3.977063901\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.977063901\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-19.7 + 60.8i)T \) |
| 11 | \( 1 + (-5.36e6 + 2.38e6i)T \) |
| good | 3 | \( 1 + (-1.76e3 + 1.28e3i)T + (4.92e5 - 1.51e6i)T^{2} \) |
| 5 | \( 1 + (-6.22e3 - 1.91e4i)T + (-9.87e8 + 7.17e8i)T^{2} \) |
| 7 | \( 1 + (-3.39e5 - 2.46e5i)T + (2.99e10 + 9.21e10i)T^{2} \) |
| 13 | \( 1 + (-1.01e7 + 3.11e7i)T + (-2.45e14 - 1.78e14i)T^{2} \) |
| 17 | \( 1 + (-1.32e7 - 4.06e7i)T + (-8.01e15 + 5.82e15i)T^{2} \) |
| 19 | \( 1 + (3.21e8 - 2.33e8i)T + (1.29e16 - 3.99e16i)T^{2} \) |
| 23 | \( 1 - 4.40e7T + 5.04e17T^{2} \) |
| 29 | \( 1 + (3.51e9 + 2.55e9i)T + (3.17e18 + 9.75e18i)T^{2} \) |
| 31 | \( 1 + (1.55e9 - 4.78e9i)T + (-1.97e19 - 1.43e19i)T^{2} \) |
| 37 | \( 1 + (-5.21e8 - 3.78e8i)T + (7.52e19 + 2.31e20i)T^{2} \) |
| 41 | \( 1 + (1.25e10 - 9.14e9i)T + (2.85e20 - 8.79e20i)T^{2} \) |
| 43 | \( 1 - 4.44e10T + 1.71e21T^{2} \) |
| 47 | \( 1 + (2.27e10 - 1.65e10i)T + (1.68e21 - 5.19e21i)T^{2} \) |
| 53 | \( 1 + (-1.48e10 + 4.57e10i)T + (-2.10e22 - 1.53e22i)T^{2} \) |
| 59 | \( 1 + (-2.39e11 - 1.73e11i)T + (3.24e22 + 9.98e22i)T^{2} \) |
| 61 | \( 1 + (-2.85e10 - 8.77e10i)T + (-1.30e23 + 9.51e22i)T^{2} \) |
| 67 | \( 1 + 8.85e11T + 5.48e23T^{2} \) |
| 71 | \( 1 + (2.03e11 + 6.25e11i)T + (-9.42e23 + 6.84e23i)T^{2} \) |
| 73 | \( 1 + (6.46e11 + 4.69e11i)T + (5.16e23 + 1.59e24i)T^{2} \) |
| 79 | \( 1 + (1.95e11 - 6.02e11i)T + (-3.77e24 - 2.74e24i)T^{2} \) |
| 83 | \( 1 + (-4.38e11 - 1.34e12i)T + (-7.17e24 + 5.21e24i)T^{2} \) |
| 89 | \( 1 - 6.72e12T + 2.19e25T^{2} \) |
| 97 | \( 1 + (6.37e11 - 1.96e12i)T + (-5.44e25 - 3.95e25i)T^{2} \) |
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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
−14.63712249490027989447610283782, −13.23458815336716003656076208446, −12.23110952060568570046594914708, −10.65035888579708989251935915356, −8.768216690107165118565765506907, −8.016278803197350484305922101981, −6.04632508579742597776334176691, −3.57905331029333680137005848404, −2.33479932148482476568195496331, −1.29347429419292090194228696198,
1.79662248816205486526869119293, 4.04094318849453729107107498240, 4.60637882307402882177475094562, 7.13947983492635428578325774797, 8.697467362236221623819642604668, 9.270251806661488698201189130655, 11.11680357953663857360394732650, 13.34462214364227124222144902693, 14.32634653437248587843559264709, 14.92307389216509250595185952963
