| L(s) = 1 | − 2-s + 2·3-s − 5·5-s − 2·6-s − 2·7-s − 8-s + 3·9-s + 5·10-s − 10·11-s + 13-s + 2·14-s − 10·15-s − 16-s − 4·17-s − 3·18-s + 4·19-s − 4·21-s + 10·22-s + 2·23-s − 2·24-s + 12·25-s − 26-s + 4·27-s − 2·29-s + 10·30-s + 6·31-s + 6·32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.15·3-s − 2.23·5-s − 0.816·6-s − 0.755·7-s − 0.353·8-s + 9-s + 1.58·10-s − 3.01·11-s + 0.277·13-s + 0.534·14-s − 2.58·15-s − 1/4·16-s − 0.970·17-s − 0.707·18-s + 0.917·19-s − 0.872·21-s + 2.13·22-s + 0.417·23-s − 0.408·24-s + 12/5·25-s − 0.196·26-s + 0.769·27-s − 0.371·29-s + 1.82·30-s + 1.07·31-s + 1.06·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 233289 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 233289 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| 23 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 + T + T^{2} + p T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + p T + 13 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 - T + 23 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 4 T + 25 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 4 T + 29 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 2 T + 7 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 + 10 T + 55 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 7 T + 17 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 10 T + 106 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 7 T + 37 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 3 T + 91 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 13 T + 135 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 13 T + 147 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 9 T + 133 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 10 T + 119 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 - 4 T + 157 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 11 T + 179 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 28 T + 377 T^{2} + 28 p T^{3} + p^{2} T^{4} \) |
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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
−10.70471667671195803796149659313, −10.39471216523477181280220242682, −9.582913285860296341262934199506, −9.577327362213233905533962947874, −8.529547507726654778529880346802, −8.522053963711516803659718828383, −8.243895622040926499752855921528, −7.76889397655228984207444622834, −7.21418436173729151984097644892, −7.08160787659260455300071561958, −6.36172370295098410355073367191, −5.40632235458926868943752543507, −4.69899128628875193921548297147, −4.63582150972919390839002698642, −3.49706935210719144215239501294, −3.10922972288336342984270277987, −3.04128306180091303932069132280, −1.94440605526149119220623336545, 0, 0,
1.94440605526149119220623336545, 3.04128306180091303932069132280, 3.10922972288336342984270277987, 3.49706935210719144215239501294, 4.63582150972919390839002698642, 4.69899128628875193921548297147, 5.40632235458926868943752543507, 6.36172370295098410355073367191, 7.08160787659260455300071561958, 7.21418436173729151984097644892, 7.76889397655228984207444622834, 8.243895622040926499752855921528, 8.522053963711516803659718828383, 8.529547507726654778529880346802, 9.577327362213233905533962947874, 9.582913285860296341262934199506, 10.39471216523477181280220242682, 10.70471667671195803796149659313
