| L(s) = 1 | + 1.09e12·2-s + 2.07e19·3-s + 1.20e24·4-s − 2.06e28·5-s + 2.28e31·6-s + 2.86e34·7-s + 1.32e36·8-s − 1.16e37·9-s − 2.27e40·10-s + 2.41e42·11-s + 2.51e43·12-s + 1.54e45·13-s + 3.14e46·14-s − 4.29e47·15-s + 1.46e48·16-s − 6.48e49·17-s − 1.28e49·18-s − 6.35e51·19-s − 2.49e52·20-s + 5.95e53·21-s + 2.65e54·22-s − 1.06e55·23-s + 2.76e55·24-s + 1.29e55·25-s + 1.70e57·26-s − 9.45e57·27-s + 3.46e58·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.986·3-s + 0.5·4-s − 1.01·5-s + 0.697·6-s + 1.70·7-s + 0.353·8-s − 0.0263·9-s − 0.718·10-s + 1.60·11-s + 0.493·12-s + 1.18·13-s + 1.20·14-s − 1.00·15-s + 0.250·16-s − 0.952·17-s − 0.0186·18-s − 1.03·19-s − 0.507·20-s + 1.67·21-s + 1.13·22-s − 0.751·23-s + 0.348·24-s + 0.0313·25-s + 0.840·26-s − 1.01·27-s + 0.850·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(82-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s+81/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(41)\) |
\(\approx\) |
\(5.705837814\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.705837814\) |
| \(L(\frac{83}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 1.09e12T \) |
| good | 3 | \( 1 - 2.07e19T + 4.43e38T^{2} \) |
| 5 | \( 1 + 2.06e28T + 4.13e56T^{2} \) |
| 7 | \( 1 - 2.86e34T + 2.83e68T^{2} \) |
| 11 | \( 1 - 2.41e42T + 2.25e84T^{2} \) |
| 13 | \( 1 - 1.54e45T + 1.69e90T^{2} \) |
| 17 | \( 1 + 6.48e49T + 4.63e99T^{2} \) |
| 19 | \( 1 + 6.35e51T + 3.79e103T^{2} \) |
| 23 | \( 1 + 1.06e55T + 1.99e110T^{2} \) |
| 29 | \( 1 - 2.92e59T + 2.84e118T^{2} \) |
| 31 | \( 1 - 2.92e60T + 6.31e120T^{2} \) |
| 37 | \( 1 - 2.04e63T + 1.05e127T^{2} \) |
| 41 | \( 1 - 6.67e64T + 4.32e130T^{2} \) |
| 43 | \( 1 - 7.42e65T + 2.04e132T^{2} \) |
| 47 | \( 1 - 3.75e66T + 2.75e135T^{2} \) |
| 53 | \( 1 + 1.94e69T + 4.63e139T^{2} \) |
| 59 | \( 1 - 5.44e71T + 2.74e143T^{2} \) |
| 61 | \( 1 + 5.75e71T + 4.08e144T^{2} \) |
| 67 | \( 1 - 5.74e73T + 8.16e147T^{2} \) |
| 71 | \( 1 - 1.33e75T + 8.95e149T^{2} \) |
| 73 | \( 1 + 7.56e74T + 8.49e150T^{2} \) |
| 79 | \( 1 + 6.60e76T + 5.10e153T^{2} \) |
| 83 | \( 1 - 4.24e77T + 2.78e155T^{2} \) |
| 89 | \( 1 - 1.16e78T + 7.95e157T^{2} \) |
| 97 | \( 1 + 1.14e80T + 8.48e160T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.83777012457182948488710290700, −11.86811860621667901556755611460, −11.12459693819472538975312251527, −8.642066739537162167075912311131, −8.064074493307377880060133356257, −6.39903658654996153532488242974, −4.39892408420480836732326441934, −3.87859343166051952528204844995, −2.30191900944389513159855027303, −1.13982943665743700207569397739,
1.13982943665743700207569397739, 2.30191900944389513159855027303, 3.87859343166051952528204844995, 4.39892408420480836732326441934, 6.39903658654996153532488242974, 8.064074493307377880060133356257, 8.642066739537162167075912311131, 11.12459693819472538975312251527, 11.86811860621667901556755611460, 13.83777012457182948488710290700
