| L(s) = 1 | + 2-s + 3-s + 4-s − 3·5-s + 6-s + 8-s + 9-s − 3·10-s − 2·11-s + 12-s − 13-s − 3·15-s + 16-s + 17-s + 18-s − 3·20-s − 2·22-s + 4·23-s + 24-s + 4·25-s − 26-s + 27-s + 29-s − 3·30-s − 9·31-s + 32-s − 2·33-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 1.34·5-s + 0.408·6-s + 0.353·8-s + 1/3·9-s − 0.948·10-s − 0.603·11-s + 0.288·12-s − 0.277·13-s − 0.774·15-s + 1/4·16-s + 0.242·17-s + 0.235·18-s − 0.670·20-s − 0.426·22-s + 0.834·23-s + 0.204·24-s + 4/5·25-s − 0.196·26-s + 0.192·27-s + 0.185·29-s − 0.547·30-s − 1.61·31-s + 0.176·32-s − 0.348·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4998 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4998 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 + 9 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 13 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 7 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.68087590117930770347833351460, −7.37580501759879526045193821007, −6.56147393784780380988161313710, −5.49512973316416628354608497287, −4.79589631849183177565744793336, −4.07607156018655154696428994817, −3.34655240270615378145590333791, −2.76617019565383577385415924657, −1.55754319593526391446868387581, 0,
1.55754319593526391446868387581, 2.76617019565383577385415924657, 3.34655240270615378145590333791, 4.07607156018655154696428994817, 4.79589631849183177565744793336, 5.49512973316416628354608497287, 6.56147393784780380988161313710, 7.37580501759879526045193821007, 7.68087590117930770347833351460
