| L(s) = 1 | + 2-s + 3-s + 4-s − 2·5-s + 6-s − 7-s + 8-s + 9-s − 2·10-s + 4·11-s + 12-s + 2·13-s − 14-s − 2·15-s + 16-s + 6·17-s + 18-s − 2·20-s − 21-s + 4·22-s + 23-s + 24-s − 25-s + 2·26-s + 27-s − 28-s − 2·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.894·5-s + 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.632·10-s + 1.20·11-s + 0.288·12-s + 0.554·13-s − 0.267·14-s − 0.516·15-s + 1/4·16-s + 1.45·17-s + 0.235·18-s − 0.447·20-s − 0.218·21-s + 0.852·22-s + 0.208·23-s + 0.204·24-s − 1/5·25-s + 0.392·26-s + 0.192·27-s − 0.188·28-s − 0.371·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.785859174\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.785859174\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−9.968594132488272489489411895531, −9.190747247293542996877454373507, −8.181256656550651102932966383722, −7.52050133794608477848856769448, −6.59428768312870858781878008825, −5.73334052480248554670358444617, −4.39376770604557496901808540228, −3.72175117496061513035390988095, −2.97798787924572985190414192753, −1.31754563204199673281891838511,
1.31754563204199673281891838511, 2.97798787924572985190414192753, 3.72175117496061513035390988095, 4.39376770604557496901808540228, 5.73334052480248554670358444617, 6.59428768312870858781878008825, 7.52050133794608477848856769448, 8.181256656550651102932966383722, 9.190747247293542996877454373507, 9.968594132488272489489411895531
