| L(s) = 1 | + 0.455·2-s + 3-s − 1.79·4-s − 1.16·5-s + 0.455·6-s + 0.0147·7-s − 1.72·8-s + 9-s − 0.530·10-s + 5.67·11-s − 1.79·12-s − 0.904·13-s + 0.00671·14-s − 1.16·15-s + 2.79·16-s + 17-s + 0.455·18-s − 2.16·19-s + 2.08·20-s + 0.0147·21-s + 2.58·22-s − 7.93·23-s − 1.72·24-s − 3.64·25-s − 0.412·26-s + 27-s − 0.0263·28-s + ⋯ |
| L(s) = 1 | + 0.322·2-s + 0.577·3-s − 0.896·4-s − 0.520·5-s + 0.186·6-s + 0.00556·7-s − 0.611·8-s + 0.333·9-s − 0.167·10-s + 1.71·11-s − 0.517·12-s − 0.250·13-s + 0.00179·14-s − 0.300·15-s + 0.699·16-s + 0.242·17-s + 0.107·18-s − 0.496·19-s + 0.466·20-s + 0.00321·21-s + 0.551·22-s − 1.65·23-s − 0.352·24-s − 0.729·25-s − 0.0808·26-s + 0.192·27-s − 0.00498·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 157 | \( 1 + T \) |
| good | 2 | \( 1 - 0.455T + 2T^{2} \) |
| 5 | \( 1 + 1.16T + 5T^{2} \) |
| 7 | \( 1 - 0.0147T + 7T^{2} \) |
| 11 | \( 1 - 5.67T + 11T^{2} \) |
| 13 | \( 1 + 0.904T + 13T^{2} \) |
| 19 | \( 1 + 2.16T + 19T^{2} \) |
| 23 | \( 1 + 7.93T + 23T^{2} \) |
| 29 | \( 1 - 7.32T + 29T^{2} \) |
| 31 | \( 1 + 7.47T + 31T^{2} \) |
| 37 | \( 1 + 7.56T + 37T^{2} \) |
| 41 | \( 1 - 3.28T + 41T^{2} \) |
| 43 | \( 1 + 4.48T + 43T^{2} \) |
| 47 | \( 1 - 9.65T + 47T^{2} \) |
| 53 | \( 1 - 1.88T + 53T^{2} \) |
| 59 | \( 1 - 3.49T + 59T^{2} \) |
| 61 | \( 1 - 5.75T + 61T^{2} \) |
| 67 | \( 1 + 1.33T + 67T^{2} \) |
| 71 | \( 1 - 7.45T + 71T^{2} \) |
| 73 | \( 1 + 9.61T + 73T^{2} \) |
| 79 | \( 1 - 12.0T + 79T^{2} \) |
| 83 | \( 1 + 0.418T + 83T^{2} \) |
| 89 | \( 1 - 3.01T + 89T^{2} \) |
| 97 | \( 1 - 4.15T + 97T^{2} \) |
| 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
−7.62123692140264692591589002548, −6.76074805173140755161842095525, −6.12583085803534641525473136605, −5.28816655264829119405958500421, −4.38612395826063988712632077402, −3.84874031037910588362342859264, −3.53629992340489765716032713392, −2.28179428496785587490929860427, −1.27741165292113458150691914475, 0,
1.27741165292113458150691914475, 2.28179428496785587490929860427, 3.53629992340489765716032713392, 3.84874031037910588362342859264, 4.38612395826063988712632077402, 5.28816655264829119405958500421, 6.12583085803534641525473136605, 6.76074805173140755161842095525, 7.62123692140264692591589002548
