| L(s) = 1 | + 2.23·2-s − 3-s + 2.99·4-s − 1.84·5-s − 2.23·6-s − 1.20·7-s + 2.22·8-s + 9-s − 4.12·10-s + 5.35·11-s − 2.99·12-s − 0.887·13-s − 2.69·14-s + 1.84·15-s − 1.01·16-s + 17-s + 2.23·18-s + 0.600·19-s − 5.53·20-s + 1.20·21-s + 11.9·22-s + 2.87·23-s − 2.22·24-s − 1.59·25-s − 1.98·26-s − 27-s − 3.61·28-s + ⋯ |
| L(s) = 1 | + 1.58·2-s − 0.577·3-s + 1.49·4-s − 0.825·5-s − 0.912·6-s − 0.455·7-s + 0.788·8-s + 0.333·9-s − 1.30·10-s + 1.61·11-s − 0.865·12-s − 0.246·13-s − 0.720·14-s + 0.476·15-s − 0.252·16-s + 0.242·17-s + 0.526·18-s + 0.137·19-s − 1.23·20-s + 0.263·21-s + 2.55·22-s + 0.599·23-s − 0.455·24-s − 0.318·25-s − 0.389·26-s − 0.192·27-s − 0.683·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.463770199\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.463770199\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 157 | \( 1 + T \) |
| good | 2 | \( 1 - 2.23T + 2T^{2} \) |
| 5 | \( 1 + 1.84T + 5T^{2} \) |
| 7 | \( 1 + 1.20T + 7T^{2} \) |
| 11 | \( 1 - 5.35T + 11T^{2} \) |
| 13 | \( 1 + 0.887T + 13T^{2} \) |
| 19 | \( 1 - 0.600T + 19T^{2} \) |
| 23 | \( 1 - 2.87T + 23T^{2} \) |
| 29 | \( 1 + 3.05T + 29T^{2} \) |
| 31 | \( 1 + 0.190T + 31T^{2} \) |
| 37 | \( 1 + 4.22T + 37T^{2} \) |
| 41 | \( 1 - 3.62T + 41T^{2} \) |
| 43 | \( 1 - 12.6T + 43T^{2} \) |
| 47 | \( 1 - 1.29T + 47T^{2} \) |
| 53 | \( 1 - 0.764T + 53T^{2} \) |
| 59 | \( 1 - 14.1T + 59T^{2} \) |
| 61 | \( 1 + 2.28T + 61T^{2} \) |
| 67 | \( 1 - 5.95T + 67T^{2} \) |
| 71 | \( 1 + 4.89T + 71T^{2} \) |
| 73 | \( 1 - 16.4T + 73T^{2} \) |
| 79 | \( 1 - 1.52T + 79T^{2} \) |
| 83 | \( 1 - 3.82T + 83T^{2} \) |
| 89 | \( 1 + 2.46T + 89T^{2} \) |
| 97 | \( 1 + 11.3T + 97T^{2} \) |
| 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
−7.33161451882506832460839083338, −6.98318027607072147185614146514, −6.26374827048093751772542626090, −5.70428598351917977048755930933, −4.96885952149316565353044487618, −4.11868338231575717101205043389, −3.84821047611731093947760450075, −3.10560283377711080100690541315, −2.00559344189623540322100616594, −0.74280901735372869535562708046,
0.74280901735372869535562708046, 2.00559344189623540322100616594, 3.10560283377711080100690541315, 3.84821047611731093947760450075, 4.11868338231575717101205043389, 4.96885952149316565353044487618, 5.70428598351917977048755930933, 6.26374827048093751772542626090, 6.98318027607072147185614146514, 7.33161451882506832460839083338
