| L(s) = 1 | + 1.87·2-s + 1.28·3-s + 2.53·4-s + 0.684·5-s + 2.41·6-s − 7-s + 2.87·8-s + 0.652·9-s + 1.28·10-s − 11-s + 3.25·12-s − 1.73·13-s − 1.87·14-s + 0.879·15-s + 2.87·16-s − 1.96·17-s + 1.22·18-s + 1.73·20-s − 1.28·21-s − 1.87·22-s + 23-s + 3.70·24-s − 0.532·25-s − 3.25·26-s − 0.446·27-s − 2.53·28-s + 1.53·29-s + ⋯ |
| L(s) = 1 | + 1.87·2-s + 1.28·3-s + 2.53·4-s + 0.684·5-s + 2.41·6-s − 7-s + 2.87·8-s + 0.652·9-s + 1.28·10-s − 11-s + 3.25·12-s − 1.73·13-s − 1.87·14-s + 0.879·15-s + 2.87·16-s − 1.96·17-s + 1.22·18-s + 1.73·20-s − 1.28·21-s − 1.87·22-s + 23-s + 3.70·24-s − 0.532·25-s − 3.25·26-s − 0.446·27-s − 2.53·28-s + 1.53·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3311 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3311 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(5.269367784\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.269367784\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 43 | \( 1 - T \) |
| good | 2 | \( 1 - 1.87T + T^{2} \) |
| 3 | \( 1 - 1.28T + T^{2} \) |
| 5 | \( 1 - 0.684T + T^{2} \) |
| 13 | \( 1 + 1.73T + T^{2} \) |
| 17 | \( 1 + 1.96T + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - T + T^{2} \) |
| 29 | \( 1 - 1.53T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - 1.96T + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - 0.347T + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + 1.87T + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - 1.28T + T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 - T^{2} \) |
| 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
−8.894989421487216382683199477508, −7.69700929506399408340021405204, −7.17787965206611912902088614782, −6.42735368737720169199084908314, −5.69235973575059459192737165855, −4.77662811109005533339389124303, −4.22750218161341761354355288077, −3.07792495138543732068890134561, −2.47600462797563496725773151650, −2.32042870900223378047411027926,
2.32042870900223378047411027926, 2.47600462797563496725773151650, 3.07792495138543732068890134561, 4.22750218161341761354355288077, 4.77662811109005533339389124303, 5.69235973575059459192737165855, 6.42735368737720169199084908314, 7.17787965206611912902088614782, 7.69700929506399408340021405204, 8.894989421487216382683199477508
