| L(s) = 1 | + 0.649·2-s − 0.817·3-s − 1.57·4-s + 5-s − 0.531·6-s − 7-s − 2.32·8-s − 2.33·9-s + 0.649·10-s − 3.00·11-s + 1.28·12-s + 2.51·13-s − 0.649·14-s − 0.817·15-s + 1.64·16-s + 1.52·17-s − 1.51·18-s − 1.20·19-s − 1.57·20-s + 0.817·21-s − 1.95·22-s + 1.20·23-s + 1.90·24-s + 25-s + 1.63·26-s + 4.35·27-s + 1.57·28-s + ⋯ |
| L(s) = 1 | + 0.459·2-s − 0.471·3-s − 0.788·4-s + 0.447·5-s − 0.216·6-s − 0.377·7-s − 0.822·8-s − 0.777·9-s + 0.205·10-s − 0.904·11-s + 0.372·12-s + 0.696·13-s − 0.173·14-s − 0.211·15-s + 0.410·16-s + 0.370·17-s − 0.357·18-s − 0.276·19-s − 0.352·20-s + 0.178·21-s − 0.415·22-s + 0.250·23-s + 0.387·24-s + 0.200·25-s + 0.320·26-s + 0.838·27-s + 0.298·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 229 | \( 1 - T \) |
| good | 2 | \( 1 - 0.649T + 2T^{2} \) |
| 3 | \( 1 + 0.817T + 3T^{2} \) |
| 11 | \( 1 + 3.00T + 11T^{2} \) |
| 13 | \( 1 - 2.51T + 13T^{2} \) |
| 17 | \( 1 - 1.52T + 17T^{2} \) |
| 19 | \( 1 + 1.20T + 19T^{2} \) |
| 23 | \( 1 - 1.20T + 23T^{2} \) |
| 29 | \( 1 - 2.37T + 29T^{2} \) |
| 31 | \( 1 - 1.08T + 31T^{2} \) |
| 37 | \( 1 + 4.97T + 37T^{2} \) |
| 41 | \( 1 - 6.52T + 41T^{2} \) |
| 43 | \( 1 - 7.05T + 43T^{2} \) |
| 47 | \( 1 - 4.89T + 47T^{2} \) |
| 53 | \( 1 + 12.1T + 53T^{2} \) |
| 59 | \( 1 - 5.15T + 59T^{2} \) |
| 61 | \( 1 + 9.88T + 61T^{2} \) |
| 67 | \( 1 - 9.48T + 67T^{2} \) |
| 71 | \( 1 + 4.00T + 71T^{2} \) |
| 73 | \( 1 + 3.65T + 73T^{2} \) |
| 79 | \( 1 - 0.880T + 79T^{2} \) |
| 83 | \( 1 + 9.03T + 83T^{2} \) |
| 89 | \( 1 - 16.5T + 89T^{2} \) |
| 97 | \( 1 - 8.77T + 97T^{2} \) |
| 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
−7.53924122132522964098628765272, −6.38890089510950359556407354215, −6.03574909266990430780757142131, −5.36299354371485281656225002578, −4.86496710536557921066744862481, −3.95992483500010442604398051757, −3.13533013908857392129911317017, −2.50427439840512583977564007483, −1.03684647770068607623659563013, 0,
1.03684647770068607623659563013, 2.50427439840512583977564007483, 3.13533013908857392129911317017, 3.95992483500010442604398051757, 4.86496710536557921066744862481, 5.36299354371485281656225002578, 6.03574909266990430780757142131, 6.38890089510950359556407354215, 7.53924122132522964098628765272
