| L(s) = 1 | − 2-s − 7-s + 8-s + 11-s − 5·13-s + 14-s − 16-s − 7·17-s + 7·19-s − 22-s + 2·23-s − 10·25-s + 5·26-s + 29-s − 4·31-s + 7·34-s + 2·37-s − 7·38-s − 9·41-s − 2·46-s − 10·47-s + 10·50-s − 22·53-s − 56-s − 58-s + 5·61-s + 4·62-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.377·7-s + 0.353·8-s + 0.301·11-s − 1.38·13-s + 0.267·14-s − 1/4·16-s − 1.69·17-s + 1.60·19-s − 0.213·22-s + 0.417·23-s − 2·25-s + 0.980·26-s + 0.185·29-s − 0.718·31-s + 1.20·34-s + 0.328·37-s − 1.13·38-s − 1.40·41-s − 0.294·46-s − 1.45·47-s + 1.41·50-s − 3.02·53-s − 0.133·56-s − 0.131·58-s + 0.640·61-s + 0.508·62-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2683044 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2683044 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T + T^{2} \) |
| 13 | $C_2$ | \( 1 + 5 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - T - 10 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 7 T + 32 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 2 T - 19 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - T - 28 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 2 T - 33 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 9 T + 40 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 5 T - 36 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 8 T - 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 12 T + 73 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 5 T - 64 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 4 T - 81 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
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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
−9.167839489313145055807587809787, −9.053072057675644522822382097408, −8.268217578905624172680115462462, −8.134521306975203985693794683150, −7.44744039826099146807680543395, −7.43845103035898628856950885745, −6.75975385022783921262740502385, −6.60767958771482439706137785431, −5.94357320545939380800533883973, −5.54264603438607734946730230880, −4.97240081273556914436296916133, −4.66887109773685991954176406148, −4.17469817271436241338481375774, −3.62354003022175483778241356222, −2.95858744066620481661029742455, −2.67956833095959005023292339785, −1.65630130364636584965460315202, −1.59661862153011370479216490860, 0, 0,
1.59661862153011370479216490860, 1.65630130364636584965460315202, 2.67956833095959005023292339785, 2.95858744066620481661029742455, 3.62354003022175483778241356222, 4.17469817271436241338481375774, 4.66887109773685991954176406148, 4.97240081273556914436296916133, 5.54264603438607734946730230880, 5.94357320545939380800533883973, 6.60767958771482439706137785431, 6.75975385022783921262740502385, 7.43845103035898628856950885745, 7.44744039826099146807680543395, 8.134521306975203985693794683150, 8.268217578905624172680115462462, 9.053072057675644522822382097408, 9.167839489313145055807587809787
