| L(s) = 1 | + 2-s + 0.0819·3-s + 4-s + 1.32·5-s + 0.0819·6-s − 2.70·7-s + 8-s − 2.99·9-s + 1.32·10-s + 1.75·11-s + 0.0819·12-s − 4.14·13-s − 2.70·14-s + 0.108·15-s + 16-s + 6.87·17-s − 2.99·18-s + 1.00·19-s + 1.32·20-s − 0.221·21-s + 1.75·22-s − 1.96·23-s + 0.0819·24-s − 3.24·25-s − 4.14·26-s − 0.491·27-s − 2.70·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.0473·3-s + 0.5·4-s + 0.593·5-s + 0.0334·6-s − 1.02·7-s + 0.353·8-s − 0.997·9-s + 0.419·10-s + 0.529·11-s + 0.0236·12-s − 1.14·13-s − 0.722·14-s + 0.0280·15-s + 0.250·16-s + 1.66·17-s − 0.705·18-s + 0.229·19-s + 0.296·20-s − 0.0483·21-s + 0.374·22-s − 0.410·23-s + 0.0167·24-s − 0.648·25-s − 0.812·26-s − 0.0945·27-s − 0.510·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.913334692\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.913334692\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 2003 | \( 1 + T \) |
| good | 3 | \( 1 - 0.0819T + 3T^{2} \) |
| 5 | \( 1 - 1.32T + 5T^{2} \) |
| 7 | \( 1 + 2.70T + 7T^{2} \) |
| 11 | \( 1 - 1.75T + 11T^{2} \) |
| 13 | \( 1 + 4.14T + 13T^{2} \) |
| 17 | \( 1 - 6.87T + 17T^{2} \) |
| 19 | \( 1 - 1.00T + 19T^{2} \) |
| 23 | \( 1 + 1.96T + 23T^{2} \) |
| 29 | \( 1 - 9.53T + 29T^{2} \) |
| 31 | \( 1 - 8.50T + 31T^{2} \) |
| 37 | \( 1 - 2.20T + 37T^{2} \) |
| 41 | \( 1 - 4.99T + 41T^{2} \) |
| 43 | \( 1 - 6.90T + 43T^{2} \) |
| 47 | \( 1 + 6.91T + 47T^{2} \) |
| 53 | \( 1 + 6.58T + 53T^{2} \) |
| 59 | \( 1 - 14.2T + 59T^{2} \) |
| 61 | \( 1 - 7.76T + 61T^{2} \) |
| 67 | \( 1 + 5.34T + 67T^{2} \) |
| 71 | \( 1 + 1.15T + 71T^{2} \) |
| 73 | \( 1 - 9.97T + 73T^{2} \) |
| 79 | \( 1 - 8.48T + 79T^{2} \) |
| 83 | \( 1 - 13.4T + 83T^{2} \) |
| 89 | \( 1 - 0.904T + 89T^{2} \) |
| 97 | \( 1 + 2.43T + 97T^{2} \) |
| 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
−8.283711892504785836104148951445, −7.71395515382444214590583564047, −6.65034065603041529337816177767, −6.20704602466867808343186077209, −5.53000235343231608491154615000, −4.79561325333460447331238862847, −3.72222687597564279081225101848, −2.94086907418339708267716836860, −2.37716462370920085230143301943, −0.870955968348789176200988650637,
0.870955968348789176200988650637, 2.37716462370920085230143301943, 2.94086907418339708267716836860, 3.72222687597564279081225101848, 4.79561325333460447331238862847, 5.53000235343231608491154615000, 6.20704602466867808343186077209, 6.65034065603041529337816177767, 7.71395515382444214590583564047, 8.283711892504785836104148951445
