| L(s) = 1 | − 3·2-s − 3·3-s + 4·4-s − 2·5-s + 9·6-s − 6·7-s − 3·8-s + 2·9-s + 6·10-s − 12·12-s − 4·13-s + 18·14-s + 6·15-s + 3·16-s − 6·17-s − 6·18-s − 19-s − 8·20-s + 18·21-s − 2·23-s + 9·24-s + 12·26-s + 6·27-s − 24·28-s − 18·30-s − 31-s − 6·32-s + ⋯ |
| L(s) = 1 | − 2.12·2-s − 1.73·3-s + 2·4-s − 0.894·5-s + 3.67·6-s − 2.26·7-s − 1.06·8-s + 2/3·9-s + 1.89·10-s − 3.46·12-s − 1.10·13-s + 4.81·14-s + 1.54·15-s + 3/4·16-s − 1.45·17-s − 1.41·18-s − 0.229·19-s − 1.78·20-s + 3.92·21-s − 0.417·23-s + 1.83·24-s + 2.35·26-s + 1.15·27-s − 4.53·28-s − 3.28·30-s − 0.179·31-s − 1.06·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6511 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6511 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 17 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 7 T + p T^{2} ) \) |
| 383 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 6 T + p T^{2} ) \) |
| good | 2 | $C_2^2$ | \( 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 3 | $D_{4}$ | \( 1 + p T + 7 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 2 T + 4 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 4 T + 11 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 23 | $D_{4}$ | \( 1 + 2 T - 10 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 28 T^{2} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + T + 5 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 2 T + 8 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 8 T + 57 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 4 T + 15 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 7 T + 51 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 3 T + 28 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 3 T + 29 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 7 T + 57 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 16 T + 183 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 14 T + 120 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + T + 72 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 2 T + 148 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 7 T + 147 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 18 T + 235 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 23 T + 325 T^{2} + 23 p T^{3} + p^{2} T^{4} \) |
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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
−17.6744203337, −17.1362107943, −16.8414958152, −16.3693928347, −16.1086724815, −15.5631704252, −15.0771557773, −14.1795589733, −13.2185741867, −12.6928931592, −12.3451413662, −11.5921134287, −11.3666034192, −10.6347779018, −10.1222024232, −9.67871797357, −9.25663446499, −8.51844064598, −7.99130698703, −6.92393463891, −6.77448356143, −6.05860128125, −5.26041614724, −4.08011090379, −2.86047668933, 0, 0,
2.86047668933, 4.08011090379, 5.26041614724, 6.05860128125, 6.77448356143, 6.92393463891, 7.99130698703, 8.51844064598, 9.25663446499, 9.67871797357, 10.1222024232, 10.6347779018, 11.3666034192, 11.5921134287, 12.3451413662, 12.6928931592, 13.2185741867, 14.1795589733, 15.0771557773, 15.5631704252, 16.1086724815, 16.3693928347, 16.8414958152, 17.1362107943, 17.6744203337
