| L(s) = 1 | − 2-s − 5-s − 7-s + 8-s − 3·9-s + 10-s − 3·11-s − 5·13-s + 14-s − 16-s − 6·17-s + 3·18-s + 4·19-s + 3·22-s − 6·23-s + 5·26-s − 6·29-s − 2·31-s + 6·34-s + 35-s − 20·37-s − 4·38-s − 40-s + 10·43-s + 3·45-s + 6·46-s − 3·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.447·5-s − 0.377·7-s + 0.353·8-s − 9-s + 0.316·10-s − 0.904·11-s − 1.38·13-s + 0.267·14-s − 1/4·16-s − 1.45·17-s + 0.707·18-s + 0.917·19-s + 0.639·22-s − 1.25·23-s + 0.980·26-s − 1.11·29-s − 0.359·31-s + 1.02·34-s + 0.169·35-s − 3.28·37-s − 0.648·38-s − 0.158·40-s + 1.52·43-s + 0.447·45-s + 0.884·46-s − 0.437·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 396900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 396900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 3 | $C_2$ | \( 1 + p T^{2} \) |
| 5 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 11 | $C_2^2$ | \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 6 T + 7 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 2 T - 27 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 10 T + 57 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 3 T - 38 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 12 T + 85 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 10 T + 39 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 2 T - 63 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - T - 78 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 9 T - 2 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 17 T + 192 T^{2} + 17 p T^{3} + p^{2} T^{4} \) |
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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
−10.25319999840113193605985791104, −9.982347663495477407311000425765, −9.432220316330549210679677317689, −9.110278112615298860805625974258, −8.675491434505950154151157744082, −8.169120215378619190584539822270, −7.87569248634689643462928193979, −7.36035092887956838632126613266, −6.85974165607232255542363724809, −6.64805201807973562484538792722, −5.51107136098992982734167394184, −5.46784927278664523162044606806, −5.04857049395821607469841694228, −3.98912654714384482663577577652, −3.92150673435308056670677185243, −2.85542466720270114539085431931, −2.49812221506416199255590816022, −1.75220651030138585254562129753, 0, 0,
1.75220651030138585254562129753, 2.49812221506416199255590816022, 2.85542466720270114539085431931, 3.92150673435308056670677185243, 3.98912654714384482663577577652, 5.04857049395821607469841694228, 5.46784927278664523162044606806, 5.51107136098992982734167394184, 6.64805201807973562484538792722, 6.85974165607232255542363724809, 7.36035092887956838632126613266, 7.87569248634689643462928193979, 8.169120215378619190584539822270, 8.675491434505950154151157744082, 9.110278112615298860805625974258, 9.432220316330549210679677317689, 9.982347663495477407311000425765, 10.25319999840113193605985791104
