| L(s) = 1 | − 2-s − 3·3-s − 4·5-s + 3·6-s − 7·7-s + 8-s + 6·9-s + 4·10-s − 2·11-s − 5·13-s + 7·14-s + 12·15-s − 16-s + 2·17-s − 6·18-s − 4·19-s + 21·21-s + 2·22-s − 6·23-s − 3·24-s + 7·25-s + 5·26-s − 9·27-s − 12·30-s − 2·31-s + 6·33-s − 2·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.73·3-s − 1.78·5-s + 1.22·6-s − 2.64·7-s + 0.353·8-s + 2·9-s + 1.26·10-s − 0.603·11-s − 1.38·13-s + 1.87·14-s + 3.09·15-s − 1/4·16-s + 0.485·17-s − 1.41·18-s − 0.917·19-s + 4.58·21-s + 0.426·22-s − 1.25·23-s − 0.612·24-s + 7/5·25-s + 0.980·26-s − 1.73·27-s − 2.19·30-s − 0.359·31-s + 1.04·33-s − 0.342·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14724 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14724 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 3 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 409 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 16 T + p T^{2} ) \) |
| good | 5 | $D_{4}$ | \( 1 + 4 T + 9 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + p T + 25 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 2 T + 16 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 5 T + 25 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 2 T + 7 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 4 T + 3 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 6 T + 32 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 32 T^{2} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 2 T + 8 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 5 T + 17 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 18 T + 157 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 6 T + 9 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 2 T + 81 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 5 T + 7 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 6 T + 22 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 9 T + 103 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 91 T^{2} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 3 T - 14 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 2 T + 107 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + T + 5 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - T + 122 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 2 T + 22 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 4 T - 25 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
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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
−16.4646033432, −16.2276755855, −15.9052030845, −15.5592576795, −15.0168659856, −14.2648115558, −13.3454473298, −12.7758510517, −12.5806653554, −12.0643200497, −11.9215474550, −10.9453772753, −10.6553110797, −10.1325392875, −9.54976028645, −9.33622017041, −8.14675658905, −7.67765613625, −7.14969013512, −6.58104108955, −6.08634667898, −5.30715393234, −4.36184812687, −3.87128856025, −2.87272043841, 0, 0,
2.87272043841, 3.87128856025, 4.36184812687, 5.30715393234, 6.08634667898, 6.58104108955, 7.14969013512, 7.67765613625, 8.14675658905, 9.33622017041, 9.54976028645, 10.1325392875, 10.6553110797, 10.9453772753, 11.9215474550, 12.0643200497, 12.5806653554, 12.7758510517, 13.3454473298, 14.2648115558, 15.0168659856, 15.5592576795, 15.9052030845, 16.2276755855, 16.4646033432
