| L(s) = 1 | − 3·2-s − 3-s + 4·4-s − 5·5-s + 3·6-s − 3·7-s − 3·8-s + 15·10-s − 11-s − 4·12-s − 8·13-s + 9·14-s + 5·15-s + 3·16-s − 7·17-s − 20·20-s + 3·21-s + 3·22-s − 23-s + 3·24-s + 16·25-s + 24·26-s + 27-s − 12·28-s − 5·29-s − 15·30-s − 2·31-s + ⋯ |
| L(s) = 1 | − 2.12·2-s − 0.577·3-s + 2·4-s − 2.23·5-s + 1.22·6-s − 1.13·7-s − 1.06·8-s + 4.74·10-s − 0.301·11-s − 1.15·12-s − 2.21·13-s + 2.40·14-s + 1.29·15-s + 3/4·16-s − 1.69·17-s − 4.47·20-s + 0.654·21-s + 0.639·22-s − 0.208·23-s + 0.612·24-s + 16/5·25-s + 4.70·26-s + 0.192·27-s − 2.26·28-s − 0.928·29-s − 2.73·30-s − 0.359·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12105 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12105 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 4 T + p T^{2} ) \) |
| 269 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 18 T + p T^{2} ) \) |
| good | 2 | $C_2^2$ | \( 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 3 T + 3 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + T - 7 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 8 T + 40 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 7 T + 25 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + T + 37 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 5 T + 11 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 2 T + 18 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 12 T + 97 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + T - 12 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 8 T + 97 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 4 T + 32 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 6 T + 5 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 5 T - 14 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 15 T + 117 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 4 T - 2 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 14 T + 122 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 79 | $D_{4}$ | \( 1 - 4 T + 2 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 3 T - 12 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 4 T + 110 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 8 T + 88 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
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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
−16.8605249741, −16.5847606635, −16.0055693656, −15.6089746560, −15.2241353352, −14.8613614398, −14.0008546228, −13.1107552545, −12.5044398146, −12.0834049943, −11.9301164714, −11.0049467027, −10.7040238757, −10.2686499607, −9.45606927939, −9.10461871940, −8.69241607037, −7.87703083308, −7.50871333277, −7.11687378578, −6.53217116859, −5.31575370302, −4.50245249168, −3.67743149898, −2.59741178601, 0, 0,
2.59741178601, 3.67743149898, 4.50245249168, 5.31575370302, 6.53217116859, 7.11687378578, 7.50871333277, 7.87703083308, 8.69241607037, 9.10461871940, 9.45606927939, 10.2686499607, 10.7040238757, 11.0049467027, 11.9301164714, 12.0834049943, 12.5044398146, 13.1107552545, 14.0008546228, 14.8613614398, 15.2241353352, 15.6089746560, 16.0055693656, 16.5847606635, 16.8605249741
