| L(s) = 1 | − 3.35e7·2-s + 2.06e12·3-s + 1.12e15·4-s − 1.04e18·5-s − 6.93e19·6-s − 3.66e21·7-s − 3.77e22·8-s + 2.12e24·9-s + 3.51e25·10-s + 1.59e25·11-s + 2.32e27·12-s + 3.56e28·13-s + 1.22e29·14-s − 2.16e30·15-s + 1.26e30·16-s − 2.66e31·17-s − 7.12e31·18-s + 5.44e32·19-s − 1.18e33·20-s − 7.57e33·21-s − 5.33e32·22-s + 9.27e34·23-s − 7.81e34·24-s + 6.54e35·25-s − 1.19e36·26-s − 6.35e34·27-s − 4.12e36·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.40·3-s + 0.5·4-s − 1.57·5-s − 0.996·6-s − 1.03·7-s − 0.353·8-s + 0.985·9-s + 1.11·10-s + 0.0442·11-s + 0.704·12-s + 1.40·13-s + 0.730·14-s − 2.21·15-s + 0.250·16-s − 1.12·17-s − 0.697·18-s + 1.34·19-s − 0.786·20-s − 1.45·21-s − 0.0312·22-s + 1.75·23-s − 0.498·24-s + 1.47·25-s − 0.991·26-s − 0.0200·27-s − 0.516·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(52-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s+51/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(26)\) |
\(\approx\) |
\(1.560387140\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.560387140\) |
| \(L(\frac{53}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 3.35e7T \) |
| good | 3 | \( 1 - 2.06e12T + 2.15e24T^{2} \) |
| 5 | \( 1 + 1.04e18T + 4.44e35T^{2} \) |
| 7 | \( 1 + 3.66e21T + 1.25e43T^{2} \) |
| 11 | \( 1 - 1.59e25T + 1.29e53T^{2} \) |
| 13 | \( 1 - 3.56e28T + 6.47e56T^{2} \) |
| 17 | \( 1 + 2.66e31T + 5.66e62T^{2} \) |
| 19 | \( 1 - 5.44e32T + 1.64e65T^{2} \) |
| 23 | \( 1 - 9.27e34T + 2.80e69T^{2} \) |
| 29 | \( 1 + 1.09e37T + 3.82e74T^{2} \) |
| 31 | \( 1 - 1.02e38T + 1.14e76T^{2} \) |
| 37 | \( 1 - 6.63e39T + 9.51e79T^{2} \) |
| 41 | \( 1 - 1.12e41T + 1.78e82T^{2} \) |
| 43 | \( 1 + 4.52e41T + 2.02e83T^{2} \) |
| 47 | \( 1 - 2.08e42T + 1.89e85T^{2} \) |
| 53 | \( 1 - 9.74e43T + 8.67e87T^{2} \) |
| 59 | \( 1 + 8.85e44T + 2.05e90T^{2} \) |
| 61 | \( 1 - 5.77e45T + 1.12e91T^{2} \) |
| 67 | \( 1 - 3.60e46T + 1.34e93T^{2} \) |
| 71 | \( 1 + 1.74e46T + 2.59e94T^{2} \) |
| 73 | \( 1 - 1.78e47T + 1.07e95T^{2} \) |
| 79 | \( 1 + 4.78e47T + 6.01e96T^{2} \) |
| 83 | \( 1 - 6.61e47T + 7.46e97T^{2} \) |
| 89 | \( 1 - 3.82e49T + 2.62e99T^{2} \) |
| 97 | \( 1 - 2.27e49T + 2.11e101T^{2} \) |
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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
−16.05623690876738057773663387696, −15.25161997816181993514924954593, −13.24293627110695804507933437389, −11.29085135824556942535751188443, −9.242516437637359376017008173208, −8.250344226489089520364911865812, −6.98009183782805832547055051403, −3.77158508222465328688833495009, −2.90900642275175821601749738228, −0.803665498385736023372000103695,
0.803665498385736023372000103695, 2.90900642275175821601749738228, 3.77158508222465328688833495009, 6.98009183782805832547055051403, 8.250344226489089520364911865812, 9.242516437637359376017008173208, 11.29085135824556942535751188443, 13.24293627110695804507933437389, 15.25161997816181993514924954593, 16.05623690876738057773663387696
