| L(s) = 1 | + 2-s − 0.0432·3-s + 4-s − 1.25·5-s − 0.0432·6-s + 4.04·7-s + 8-s − 2.99·9-s − 1.25·10-s + 5.76·11-s − 0.0432·12-s − 0.269·13-s + 4.04·14-s + 0.0542·15-s + 16-s + 2.76·17-s − 2.99·18-s + 8.25·19-s − 1.25·20-s − 0.175·21-s + 5.76·22-s + 3.95·23-s − 0.0432·24-s − 3.42·25-s − 0.269·26-s + 0.259·27-s + 4.04·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.0249·3-s + 0.5·4-s − 0.560·5-s − 0.0176·6-s + 1.52·7-s + 0.353·8-s − 0.999·9-s − 0.396·10-s + 1.73·11-s − 0.0124·12-s − 0.0747·13-s + 1.08·14-s + 0.0140·15-s + 0.250·16-s + 0.669·17-s − 0.706·18-s + 1.89·19-s − 0.280·20-s − 0.0382·21-s + 1.22·22-s + 0.824·23-s − 0.00883·24-s − 0.685·25-s − 0.0528·26-s + 0.0499·27-s + 0.764·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.998965697\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.998965697\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 31 | \( 1 - T \) |
| 97 | \( 1 - T \) |
| good | 3 | \( 1 + 0.0432T + 3T^{2} \) |
| 5 | \( 1 + 1.25T + 5T^{2} \) |
| 7 | \( 1 - 4.04T + 7T^{2} \) |
| 11 | \( 1 - 5.76T + 11T^{2} \) |
| 13 | \( 1 + 0.269T + 13T^{2} \) |
| 17 | \( 1 - 2.76T + 17T^{2} \) |
| 19 | \( 1 - 8.25T + 19T^{2} \) |
| 23 | \( 1 - 3.95T + 23T^{2} \) |
| 29 | \( 1 - 5.26T + 29T^{2} \) |
| 37 | \( 1 + 7.47T + 37T^{2} \) |
| 41 | \( 1 + 3.75T + 41T^{2} \) |
| 43 | \( 1 + 5.56T + 43T^{2} \) |
| 47 | \( 1 + 6.13T + 47T^{2} \) |
| 53 | \( 1 + 1.80T + 53T^{2} \) |
| 59 | \( 1 - 0.642T + 59T^{2} \) |
| 61 | \( 1 + 5.60T + 61T^{2} \) |
| 67 | \( 1 - 0.508T + 67T^{2} \) |
| 71 | \( 1 + 6.25T + 71T^{2} \) |
| 73 | \( 1 - 5.29T + 73T^{2} \) |
| 79 | \( 1 - 4.93T + 79T^{2} \) |
| 83 | \( 1 + 9.19T + 83T^{2} \) |
| 89 | \( 1 - 0.00470T + 89T^{2} \) |
| 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
−8.038089483869679412398698657005, −7.34894646631122155488156092230, −6.66226912334107918840712712778, −5.75888005192871173612070164296, −5.09519232599826443795013677714, −4.58407602938999782685894318010, −3.55907560494842421793299271630, −3.13656909832862629393051717516, −1.76598162629399568964436445505, −1.05021613441045171502363227548,
1.05021613441045171502363227548, 1.76598162629399568964436445505, 3.13656909832862629393051717516, 3.55907560494842421793299271630, 4.58407602938999782685894318010, 5.09519232599826443795013677714, 5.75888005192871173612070164296, 6.66226912334107918840712712778, 7.34894646631122155488156092230, 8.038089483869679412398698657005
