L(s) = 1 | + 2-s − 2·3-s + 4-s − 2·6-s − 7-s + 8-s + 9-s + 4·11-s − 2·12-s − 2·13-s − 14-s + 16-s + 4·17-s + 18-s + 2·21-s + 4·22-s + 23-s − 2·24-s − 5·25-s − 2·26-s + 4·27-s − 28-s − 29-s − 10·31-s + 32-s − 8·33-s + 4·34-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.15·3-s + 1/2·4-s − 0.816·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s + 1.20·11-s − 0.577·12-s − 0.554·13-s − 0.267·14-s + 1/4·16-s + 0.970·17-s + 0.235·18-s + 0.436·21-s + 0.852·22-s + 0.208·23-s − 0.408·24-s − 25-s − 0.392·26-s + 0.769·27-s − 0.188·28-s − 0.185·29-s − 1.79·31-s + 0.176·32-s − 1.39·33-s + 0.685·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9338 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + T \) |
good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
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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
−7.04702318268006394021919158000, −6.58145978867175709899295156507, −5.84542791087276500355197405296, −5.41899955045890602081420861549, −4.75632996696133313983499645152, −3.80336946502404333745381531113, −3.37129921117625599375688138710, −2.15383403670028212753028308774, −1.20919625717538211058601405515, 0,
1.20919625717538211058601405515, 2.15383403670028212753028308774, 3.37129921117625599375688138710, 3.80336946502404333745381531113, 4.75632996696133313983499645152, 5.41899955045890602081420861549, 5.84542791087276500355197405296, 6.58145978867175709899295156507, 7.04702318268006394021919158000