| L(s) = 1 | − 2-s − 3-s − 4-s − 3·5-s + 6-s − 3·7-s + 3·8-s − 2·9-s + 3·10-s + 11-s + 12-s − 4·13-s + 3·14-s + 3·15-s − 16-s + 2·18-s − 3·19-s + 3·20-s + 3·21-s − 22-s − 4·23-s − 3·24-s + 4·25-s + 4·26-s + 2·27-s + 3·28-s + 29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s − 1/2·4-s − 1.34·5-s + 0.408·6-s − 1.13·7-s + 1.06·8-s − 2/3·9-s + 0.948·10-s + 0.301·11-s + 0.288·12-s − 1.10·13-s + 0.801·14-s + 0.774·15-s − 1/4·16-s + 0.471·18-s − 0.688·19-s + 0.670·20-s + 0.654·21-s − 0.213·22-s − 0.834·23-s − 0.612·24-s + 4/5·25-s + 0.784·26-s + 0.384·27-s + 0.566·28-s + 0.185·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 93952 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93952 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| 367 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 12 T + p T^{2} ) \) |
| good | 3 | $D_{4}$ | \( 1 + T + p T^{2} + p T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 3 T + p T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 3 T + 15 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - T - 14 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 3 T - 3 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 4 T + 26 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - T + 20 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 9 T + 54 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 15 T + 122 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $D_{4}$ | \( 1 + T + 25 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 5 T + 74 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 6 T + 124 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 5 T - 10 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 8 T + 110 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 5 T + 30 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 3 T + 40 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 9 T + 108 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 19 T + 218 T^{2} - 19 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 3 T + 121 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 6 T - 40 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
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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
−14.5251234273, −14.1959078129, −13.7019697780, −13.0640518314, −12.7086220381, −12.2724536267, −11.8679464581, −11.6017486424, −10.8022048264, −10.5870384446, −10.1529386387, −9.44508099495, −9.19191329690, −8.69063848644, −8.10330145926, −7.76808539378, −7.10224434001, −6.80968312837, −6.08029208060, −5.41218518578, −4.90253559492, −4.18912822945, −3.66444451165, −3.08392037265, −1.86763098992, 0, 0,
1.86763098992, 3.08392037265, 3.66444451165, 4.18912822945, 4.90253559492, 5.41218518578, 6.08029208060, 6.80968312837, 7.10224434001, 7.76808539378, 8.10330145926, 8.69063848644, 9.19191329690, 9.44508099495, 10.1529386387, 10.5870384446, 10.8022048264, 11.6017486424, 11.8679464581, 12.2724536267, 12.7086220381, 13.0640518314, 13.7019697780, 14.1959078129, 14.5251234273
