| L(s) = 1 | − 2·2-s − 3·3-s + 3·4-s + 5-s + 6·6-s − 4·7-s − 4·8-s + 6·9-s − 2·10-s + 4·11-s − 9·12-s − 4·13-s + 8·14-s − 3·15-s + 5·16-s − 4·17-s − 12·18-s − 2·19-s + 3·20-s + 12·21-s − 8·22-s + 23-s + 12·24-s + 8·26-s − 9·27-s − 12·28-s − 6·29-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.73·3-s + 3/2·4-s + 0.447·5-s + 2.44·6-s − 1.51·7-s − 1.41·8-s + 2·9-s − 0.632·10-s + 1.20·11-s − 2.59·12-s − 1.10·13-s + 2.13·14-s − 0.774·15-s + 5/4·16-s − 0.970·17-s − 2.82·18-s − 0.458·19-s + 0.670·20-s + 2.61·21-s − 1.70·22-s + 0.208·23-s + 2.44·24-s + 1.56·26-s − 1.73·27-s − 2.26·28-s − 1.11·29-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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 5 | $C_2$ | \( 1 - T + T^{2} \) |
| 7 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 11 | $C_2^2$ | \( 1 - 4 T + 5 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 4 T + 3 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 4 T - T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - T - 22 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 6 T + 7 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 10 T + 59 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 3 T - 34 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 6 T - 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 14 T + 123 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 12 T + 61 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + T - 88 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 10 T + 3 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
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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
−10.33512930763482301285433164342, −10.00062353608469862501635957660, −9.497845923632060049865465522062, −9.183093756932288274538789540247, −9.042294803804864084032562136511, −8.279515172345285294074953075947, −7.47284713122457727267891949923, −7.19526906577126156365504043438, −6.62571340683324821487931878187, −6.60481914812154171274920850709, −5.97084308352458390362534353766, −5.75249642250506811628601377933, −4.83220708416201680814919173199, −4.54383236626416990116090455562, −3.50961356314864024846800699743, −3.10960357396365457400171595436, −1.89756787697600957843629335775, −1.57410589173899393421346776134, 0, 0,
1.57410589173899393421346776134, 1.89756787697600957843629335775, 3.10960357396365457400171595436, 3.50961356314864024846800699743, 4.54383236626416990116090455562, 4.83220708416201680814919173199, 5.75249642250506811628601377933, 5.97084308352458390362534353766, 6.60481914812154171274920850709, 6.62571340683324821487931878187, 7.19526906577126156365504043438, 7.47284713122457727267891949923, 8.279515172345285294074953075947, 9.042294803804864084032562136511, 9.183093756932288274538789540247, 9.497845923632060049865465522062, 10.00062353608469862501635957660, 10.33512930763482301285433164342
