| L(s) = 1 | + 2-s + 3-s + 4-s − 5-s + 6-s − 4·7-s + 8-s + 9-s − 10-s + 11-s + 12-s − 2·13-s − 4·14-s − 15-s + 16-s + 17-s + 18-s − 20-s − 4·21-s + 22-s + 24-s + 25-s − 2·26-s + 27-s − 4·28-s + 2·29-s − 30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s − 1.51·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.301·11-s + 0.288·12-s − 0.554·13-s − 1.06·14-s − 0.258·15-s + 1/4·16-s + 0.242·17-s + 0.235·18-s − 0.223·20-s − 0.872·21-s + 0.213·22-s + 0.204·24-s + 1/5·25-s − 0.392·26-s + 0.192·27-s − 0.755·28-s + 0.371·29-s − 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5610 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5610 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 16 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−7.62899276001354048755390790523, −6.96466956198473763780310856307, −6.44579221474075152838731645611, −5.67047133947430589610536021516, −4.66575223086937146744264082264, −4.02039349226618259148360221422, −3.09754279639751207311348478175, −2.88375825008546771789056213220, −1.53801409109346474988478624537, 0,
1.53801409109346474988478624537, 2.88375825008546771789056213220, 3.09754279639751207311348478175, 4.02039349226618259148360221422, 4.66575223086937146744264082264, 5.67047133947430589610536021516, 6.44579221474075152838731645611, 6.96466956198473763780310856307, 7.62899276001354048755390790523
