| L(s) = 1 | + 2·2-s + 3-s + 3·4-s − 2·5-s + 2·6-s − 5·7-s + 4·8-s − 4·10-s − 2·11-s + 3·12-s − 8·13-s − 10·14-s − 2·15-s + 5·16-s + 17-s + 5·19-s − 6·20-s − 5·21-s − 4·22-s + 8·23-s + 4·24-s + 3·25-s − 16·26-s + 2·27-s − 15·28-s − 6·29-s − 4·30-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 0.577·3-s + 3/2·4-s − 0.894·5-s + 0.816·6-s − 1.88·7-s + 1.41·8-s − 1.26·10-s − 0.603·11-s + 0.866·12-s − 2.21·13-s − 2.67·14-s − 0.516·15-s + 5/4·16-s + 0.242·17-s + 1.14·19-s − 1.34·20-s − 1.09·21-s − 0.852·22-s + 1.66·23-s + 0.816·24-s + 3/5·25-s − 3.13·26-s + 0.384·27-s − 2.83·28-s − 1.11·29-s − 0.730·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22372900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22372900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| 43 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 - T + T^{2} - p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 5 T + 15 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 - T + 29 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 5 T + 39 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 + 6 T + 46 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 - 2 T + 62 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 3 T + 91 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + T + 59 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 13 T + 113 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 10 T + 126 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 + 27 T + 319 T^{2} + 27 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 6 T + 134 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 13 T + 195 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 9 T + 181 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 6 T - 2 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 110 T^{2} + p^{2} T^{4} \) |
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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
−7.65647965761702899428799478383, −7.52261222821761994632953868009, −7.32382913347646746933146485333, −7.23965412833762000021554283582, −6.69370993265518479142309502954, −6.32657299625521035185482852448, −5.73496060295302848807772345961, −5.52756883086344331456292602490, −5.11098750767481737116430186050, −4.73506967950212873636636449038, −4.43074698960512313983029062856, −3.86083722569694019086311466918, −3.31641819575826756251154404578, −3.24617359559533240787489824226, −2.80799156813349123919622232100, −2.70317003241382627468291642835, −1.92137718151036038199581533089, −1.26238080130978909633340299968, 0, 0,
1.26238080130978909633340299968, 1.92137718151036038199581533089, 2.70317003241382627468291642835, 2.80799156813349123919622232100, 3.24617359559533240787489824226, 3.31641819575826756251154404578, 3.86083722569694019086311466918, 4.43074698960512313983029062856, 4.73506967950212873636636449038, 5.11098750767481737116430186050, 5.52756883086344331456292602490, 5.73496060295302848807772345961, 6.32657299625521035185482852448, 6.69370993265518479142309502954, 7.23965412833762000021554283582, 7.32382913347646746933146485333, 7.52261222821761994632953868009, 7.65647965761702899428799478383
