| L(s) = 1 | + 0.176·3-s + 2.93·5-s − 0.0214·7-s − 2.96·9-s + 6.29·11-s − 2.27·13-s + 0.518·15-s + 4.51·17-s + 7.58·19-s − 0.00379·21-s − 1.87·23-s + 3.62·25-s − 1.05·27-s + 2.67·29-s + 4.43·31-s + 1.11·33-s − 0.0630·35-s − 1.65·37-s − 0.401·39-s + 2.00·41-s − 11.7·43-s − 8.71·45-s − 9.75·47-s − 6.99·49-s + 0.797·51-s + 4.87·53-s + 18.4·55-s + ⋯ |
| L(s) = 1 | + 0.101·3-s + 1.31·5-s − 0.00811·7-s − 0.989·9-s + 1.89·11-s − 0.630·13-s + 0.133·15-s + 1.09·17-s + 1.73·19-s − 0.000827·21-s − 0.391·23-s + 0.724·25-s − 0.202·27-s + 0.496·29-s + 0.797·31-s + 0.193·33-s − 0.0106·35-s − 0.271·37-s − 0.0643·39-s + 0.313·41-s − 1.79·43-s − 1.29·45-s − 1.42·47-s − 0.999·49-s + 0.111·51-s + 0.669·53-s + 2.49·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.879009423\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.879009423\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 503 | \( 1 + T \) |
| good | 3 | \( 1 - 0.176T + 3T^{2} \) |
| 5 | \( 1 - 2.93T + 5T^{2} \) |
| 7 | \( 1 + 0.0214T + 7T^{2} \) |
| 11 | \( 1 - 6.29T + 11T^{2} \) |
| 13 | \( 1 + 2.27T + 13T^{2} \) |
| 17 | \( 1 - 4.51T + 17T^{2} \) |
| 19 | \( 1 - 7.58T + 19T^{2} \) |
| 23 | \( 1 + 1.87T + 23T^{2} \) |
| 29 | \( 1 - 2.67T + 29T^{2} \) |
| 31 | \( 1 - 4.43T + 31T^{2} \) |
| 37 | \( 1 + 1.65T + 37T^{2} \) |
| 41 | \( 1 - 2.00T + 41T^{2} \) |
| 43 | \( 1 + 11.7T + 43T^{2} \) |
| 47 | \( 1 + 9.75T + 47T^{2} \) |
| 53 | \( 1 - 4.87T + 53T^{2} \) |
| 59 | \( 1 + 11.3T + 59T^{2} \) |
| 61 | \( 1 - 10.5T + 61T^{2} \) |
| 67 | \( 1 - 13.0T + 67T^{2} \) |
| 71 | \( 1 - 15.5T + 71T^{2} \) |
| 73 | \( 1 + 10.7T + 73T^{2} \) |
| 79 | \( 1 - 3.46T + 79T^{2} \) |
| 83 | \( 1 + 12.3T + 83T^{2} \) |
| 89 | \( 1 - 15.0T + 89T^{2} \) |
| 97 | \( 1 - 11.5T + 97T^{2} \) |
| 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
−8.519363605492580650258197987205, −7.79421994950515568529872374289, −6.71728691910560000085541473131, −6.30483917667409420894668604845, −5.46324146267713920238306461725, −4.94697441090499881696128540636, −3.61559245453466731027459370111, −2.99824621961325673109766035123, −1.88560099205201334692781631099, −1.04545213168128251201206409471,
1.04545213168128251201206409471, 1.88560099205201334692781631099, 2.99824621961325673109766035123, 3.61559245453466731027459370111, 4.94697441090499881696128540636, 5.46324146267713920238306461725, 6.30483917667409420894668604845, 6.71728691910560000085541473131, 7.79421994950515568529872374289, 8.519363605492580650258197987205
