| L(s) = 1 | − 2·3-s + 4-s + 2·7-s + 9-s − 2·12-s + 16-s + 12·17-s + 4·19-s − 4·21-s − 10·25-s + 4·27-s + 2·28-s − 12·29-s − 4·31-s + 36-s − 2·48-s + 3·49-s − 24·51-s + 12·53-s − 8·57-s + 2·63-s + 64-s − 8·67-s + 12·68-s + 20·75-s + 4·76-s − 11·81-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1/2·4-s + 0.755·7-s + 1/3·9-s − 0.577·12-s + 1/4·16-s + 2.91·17-s + 0.917·19-s − 0.872·21-s − 2·25-s + 0.769·27-s + 0.377·28-s − 2.22·29-s − 0.718·31-s + 1/6·36-s − 0.288·48-s + 3/7·49-s − 3.36·51-s + 1.64·53-s − 1.05·57-s + 0.251·63-s + 1/8·64-s − 0.977·67-s + 1.45·68-s + 2.30·75-s + 0.458·76-s − 1.22·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1695204 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1695204 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| 31 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
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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
−7.57571100088867902110310233811, −7.30736703591147629891449858677, −6.87864980151121342962348950105, −6.06333025434356752156931321844, −5.64737274576622310994707396282, −5.57928681742950427486583645839, −5.40092555218885150177252880861, −4.66436777849708723827288400925, −3.87479063592422741555464845270, −3.69790693414722776307982968980, −3.04217899155351164880672408498, −2.32723227474234516009046308721, −1.49056098714278580820712650479, −1.20407232976019650888554434791, 0,
1.20407232976019650888554434791, 1.49056098714278580820712650479, 2.32723227474234516009046308721, 3.04217899155351164880672408498, 3.69790693414722776307982968980, 3.87479063592422741555464845270, 4.66436777849708723827288400925, 5.40092555218885150177252880861, 5.57928681742950427486583645839, 5.64737274576622310994707396282, 6.06333025434356752156931321844, 6.87864980151121342962348950105, 7.30736703591147629891449858677, 7.57571100088867902110310233811
