| L(s) = 1 | + (45.2 + 78.3i)2-s + (−262. − 151. i)3-s + (−4.09e3 + 7.09e3i)4-s + (1.06e5 − 6.13e4i)5-s − 2.74e4i·6-s + (−7.89e5 + 2.33e5i)7-s − 7.41e5·8-s + (−2.34e6 − 4.06e6i)9-s + (9.61e6 + 5.54e6i)10-s + (9.98e6 − 1.73e7i)11-s + (2.14e6 − 1.24e6i)12-s − 4.33e7i·13-s + (−5.40e7 − 5.13e7i)14-s − 3.71e7·15-s + (−3.35e7 − 5.81e7i)16-s + (4.48e8 + 2.59e8i)17-s + ⋯ |
| L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.119 − 0.0692i)3-s + (−0.249 + 0.433i)4-s + (1.35 − 0.784i)5-s − 0.0979i·6-s + (−0.958 + 0.283i)7-s − 0.353·8-s + (−0.490 − 0.849i)9-s + (0.961 + 0.554i)10-s + (0.512 − 0.887i)11-s + (0.0599 − 0.0346i)12-s − 0.690i·13-s + (−0.512 − 0.486i)14-s − 0.217·15-s + (−0.125 − 0.216i)16-s + (1.09 + 0.631i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.806 + 0.591i)\, \overline{\Lambda}(15-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+7) \, L(s)\cr =\mathstrut & (0.806 + 0.591i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{15}{2})\) |
\(\approx\) |
\(2.08744 - 0.683342i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.08744 - 0.683342i\) |
| \(L(8)\) |
|
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 | \( 1 + (7.89e5 - 2.33e5i)T \) |
| good | 3 | \( 1 + (262. + 151. i)T + (2.39e6 + 4.14e6i)T^{2} \) |
| 5 | \( 1 + (-1.06e5 + 6.13e4i)T + (3.05e9 - 5.28e9i)T^{2} \) |
| 11 | \( 1 + (-9.98e6 + 1.73e7i)T + (-1.89e14 - 3.28e14i)T^{2} \) |
| 13 | \( 1 + 4.33e7iT - 3.93e15T^{2} \) |
| 17 | \( 1 + (-4.48e8 - 2.59e8i)T + (8.41e16 + 1.45e17i)T^{2} \) |
| 19 | \( 1 + (-8.44e8 + 4.87e8i)T + (3.99e17 - 6.91e17i)T^{2} \) |
| 23 | \( 1 + (1.28e9 + 2.22e9i)T + (-5.79e18 + 1.00e19i)T^{2} \) |
| 29 | \( 1 - 3.33e10T + 2.97e20T^{2} \) |
| 31 | \( 1 + (3.94e10 + 2.28e10i)T + (3.78e20 + 6.55e20i)T^{2} \) |
| 37 | \( 1 + (4.05e10 + 7.02e10i)T + (-4.50e21 + 7.80e21i)T^{2} \) |
| 41 | \( 1 - 2.23e11iT - 3.79e22T^{2} \) |
| 43 | \( 1 + 3.64e11T + 7.38e22T^{2} \) |
| 47 | \( 1 + (3.54e10 - 2.04e10i)T + (1.28e23 - 2.22e23i)T^{2} \) |
| 53 | \( 1 + (4.00e11 - 6.93e11i)T + (-6.89e23 - 1.19e24i)T^{2} \) |
| 59 | \( 1 + (-8.72e11 - 5.03e11i)T + (3.09e24 + 5.36e24i)T^{2} \) |
| 61 | \( 1 + (-1.17e12 + 6.79e11i)T + (4.93e24 - 8.55e24i)T^{2} \) |
| 67 | \( 1 + (2.43e12 - 4.21e12i)T + (-1.83e25 - 3.18e25i)T^{2} \) |
| 71 | \( 1 - 4.67e12T + 8.27e25T^{2} \) |
| 73 | \( 1 + (4.73e12 + 2.73e12i)T + (6.10e25 + 1.05e26i)T^{2} \) |
| 79 | \( 1 + (1.42e12 + 2.47e12i)T + (-1.84e26 + 3.19e26i)T^{2} \) |
| 83 | \( 1 - 5.45e12iT - 7.36e26T^{2} \) |
| 89 | \( 1 + (-3.25e13 + 1.88e13i)T + (9.78e26 - 1.69e27i)T^{2} \) |
| 97 | \( 1 + 1.14e14iT - 6.52e27T^{2} \) |
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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
−16.17892956031824383882148179339, −14.46679183018805551643348129756, −13.27977500571975310705678666412, −12.22471894411661614776141787409, −9.795262213763188434704676334647, −8.683879540715653882682560793617, −6.29563474721768753418643774608, −5.55148296885712925003341985314, −3.18857047537674541351544492693, −0.78574526028332738067867616311,
1.73340714952662175673632930977, 3.16399043239238567835351305494, 5.37203191593862222680819382431, 6.82435257811998286729964143579, 9.606599631511113251128056558259, 10.32135509888024860728449567575, 12.03875645959260545728394808838, 13.70244090926230809274363172752, 14.27248441734268208139744423112, 16.36101509139734290906343426221
