| L(s) = 1 | − 2·2-s + 2·4-s − 2·5-s − 9-s + 4·10-s + 2·13-s − 4·16-s − 8·17-s + 2·18-s − 4·20-s − 3·25-s − 4·26-s − 10·29-s + 8·32-s + 16·34-s − 2·36-s − 8·37-s − 4·41-s + 2·45-s + 49-s + 6·50-s + 4·52-s − 10·53-s + 20·58-s − 12·61-s − 8·64-s − 4·65-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s − 0.894·5-s − 1/3·9-s + 1.26·10-s + 0.554·13-s − 16-s − 1.94·17-s + 0.471·18-s − 0.894·20-s − 3/5·25-s − 0.784·26-s − 1.85·29-s + 1.41·32-s + 2.74·34-s − 1/3·36-s − 1.31·37-s − 0.624·41-s + 0.298·45-s + 1/7·49-s + 0.848·50-s + 0.554·52-s − 1.37·53-s + 2.62·58-s − 1.53·61-s − 64-s − 0.496·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1192464 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1192464 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 13 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 + 25 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 5 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 7 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 67 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 73 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 110 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 51 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 151 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 5 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.69519394202618448007204550633, −7.23966685543565994596268384951, −7.03092845601429042649054713264, −6.37880564536158778802948162237, −6.08438128655268018804505437264, −5.39727176441123470831132457412, −4.82256444477867172068844542846, −4.32524747834379556749989773300, −3.87042173203129843101493213742, −3.37609454946844954285086937315, −2.59576257942983043796575068904, −1.90904621969199088972348353024, −1.46064780616025386946068472796, 0, 0,
1.46064780616025386946068472796, 1.90904621969199088972348353024, 2.59576257942983043796575068904, 3.37609454946844954285086937315, 3.87042173203129843101493213742, 4.32524747834379556749989773300, 4.82256444477867172068844542846, 5.39727176441123470831132457412, 6.08438128655268018804505437264, 6.37880564536158778802948162237, 7.03092845601429042649054713264, 7.23966685543565994596268384951, 7.69519394202618448007204550633
