| L(s) = 1 | + (45.2 − 78.3i)2-s + (−1.56e3 + 902. i)3-s + (−4.09e3 − 7.09e3i)4-s + (2.52e4 + 1.45e4i)5-s + 1.63e5i·6-s + (8.23e5 − 2.49e4i)7-s − 7.41e5·8-s + (−7.61e5 + 1.31e6i)9-s + (2.28e6 − 1.31e6i)10-s + (−6.64e6 − 1.15e7i)11-s + (1.28e7 + 7.39e6i)12-s − 1.05e8i·13-s + (3.52e7 − 6.56e7i)14-s − 5.25e7·15-s + (−3.35e7 + 5.81e7i)16-s + (−7.88e7 + 4.55e7i)17-s + ⋯ |
| L(s) = 1 | + (0.353 − 0.612i)2-s + (−0.714 + 0.412i)3-s + (−0.249 − 0.433i)4-s + (0.322 + 0.186i)5-s + 0.583i·6-s + (0.999 − 0.0302i)7-s − 0.353·8-s + (−0.159 + 0.275i)9-s + (0.228 − 0.131i)10-s + (−0.340 − 0.590i)11-s + (0.357 + 0.206i)12-s − 1.68i·13-s + (0.334 − 0.622i)14-s − 0.307·15-s + (−0.125 + 0.216i)16-s + (−0.192 + 0.110i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.581 + 0.813i)\, \overline{\Lambda}(15-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+7) \, L(s)\cr =\mathstrut & (-0.581 + 0.813i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{15}{2})\) |
\(\approx\) |
\(0.606711 - 1.17867i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.606711 - 1.17867i\) |
| \(L(8)\) |
|
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 | \( 1 + (-8.23e5 + 2.49e4i)T \) |
| good | 3 | \( 1 + (1.56e3 - 902. i)T + (2.39e6 - 4.14e6i)T^{2} \) |
| 5 | \( 1 + (-2.52e4 - 1.45e4i)T + (3.05e9 + 5.28e9i)T^{2} \) |
| 11 | \( 1 + (6.64e6 + 1.15e7i)T + (-1.89e14 + 3.28e14i)T^{2} \) |
| 13 | \( 1 + 1.05e8iT - 3.93e15T^{2} \) |
| 17 | \( 1 + (7.88e7 - 4.55e7i)T + (8.41e16 - 1.45e17i)T^{2} \) |
| 19 | \( 1 + (1.00e9 + 5.77e8i)T + (3.99e17 + 6.91e17i)T^{2} \) |
| 23 | \( 1 + (-1.78e9 + 3.08e9i)T + (-5.79e18 - 1.00e19i)T^{2} \) |
| 29 | \( 1 - 1.10e10T + 2.97e20T^{2} \) |
| 31 | \( 1 + (1.63e10 - 9.43e9i)T + (3.78e20 - 6.55e20i)T^{2} \) |
| 37 | \( 1 + (-1.74e10 + 3.02e10i)T + (-4.50e21 - 7.80e21i)T^{2} \) |
| 41 | \( 1 - 7.88e9iT - 3.79e22T^{2} \) |
| 43 | \( 1 - 2.96e11T + 7.38e22T^{2} \) |
| 47 | \( 1 + (-4.20e11 - 2.42e11i)T + (1.28e23 + 2.22e23i)T^{2} \) |
| 53 | \( 1 + (9.70e11 + 1.68e12i)T + (-6.89e23 + 1.19e24i)T^{2} \) |
| 59 | \( 1 + (2.21e12 - 1.27e12i)T + (3.09e24 - 5.36e24i)T^{2} \) |
| 61 | \( 1 + (2.00e12 + 1.15e12i)T + (4.93e24 + 8.55e24i)T^{2} \) |
| 67 | \( 1 + (1.67e12 + 2.90e12i)T + (-1.83e25 + 3.18e25i)T^{2} \) |
| 71 | \( 1 - 7.07e12T + 8.27e25T^{2} \) |
| 73 | \( 1 + (1.12e12 - 6.49e11i)T + (6.10e25 - 1.05e26i)T^{2} \) |
| 79 | \( 1 + (1.76e13 - 3.05e13i)T + (-1.84e26 - 3.19e26i)T^{2} \) |
| 83 | \( 1 - 7.98e12iT - 7.36e26T^{2} \) |
| 89 | \( 1 + (6.27e13 + 3.62e13i)T + (9.78e26 + 1.69e27i)T^{2} \) |
| 97 | \( 1 + 1.90e11iT - 6.52e27T^{2} \) |
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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
−15.52019403636874953628304741314, −14.15111526419626757793899992284, −12.70351029331305036303428506662, −11.01767237568209900712368221729, −10.52550076719424799074198894511, −8.313633433321797843830708536239, −5.82404325856937605768496672088, −4.67285277799440489648791502433, −2.53543342547992782440943768853, −0.49202539503614101735372906570,
1.68270049384606095610651839271, 4.46146819459920292753419758281, 5.89074360354045949093241970017, 7.28713331016689696710973668619, 9.058753425134394032012985622238, 11.26758398428801656947062987034, 12.41549733529491331221988266086, 13.93093304828866375799856864723, 15.13536140349008869562328041750, 16.85718849781473044257896421110
