| L(s) = 1 | − 2-s + 3·3-s + 2·4-s + 6·5-s − 3·6-s − 5·8-s + 3·9-s − 6·10-s + 3·11-s + 6·12-s − 2·13-s + 18·15-s + 5·16-s + 2·17-s − 3·18-s + 19-s + 12·20-s − 3·22-s − 15·24-s + 17·25-s + 2·26-s − 7·29-s − 18·30-s + 6·31-s − 10·32-s + 9·33-s − 2·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.73·3-s + 4-s + 2.68·5-s − 1.22·6-s − 1.76·8-s + 9-s − 1.89·10-s + 0.904·11-s + 1.73·12-s − 0.554·13-s + 4.64·15-s + 5/4·16-s + 0.485·17-s − 0.707·18-s + 0.229·19-s + 2.68·20-s − 0.639·22-s − 3.06·24-s + 17/5·25-s + 0.392·26-s − 1.29·29-s − 3.28·30-s + 1.07·31-s − 1.76·32-s + 1.56·33-s − 0.342·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405769 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405769 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.397856452\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.397856452\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | | \( 1 \) |
| 13 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + T - T^{2} + p T^{3} + p^{2} T^{4} \) |
| 3 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 2 T - 13 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 7 T + 20 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 2 T - 33 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 3 T - 32 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 7 T + 6 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 4 T - 43 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 3 T - 58 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 13 T + 98 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 6 T - 53 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 19 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−10.27962439364336452354976674814, −10.16951030776102695258004177625, −9.842038728476668612465377262783, −9.324793331059242290795139831263, −9.188914032787288603776484306400, −8.699385849675814534459482604763, −8.598070929724526489489679568355, −7.84105277319775683298564600351, −7.09393507978738512689455294188, −7.07219297765751453787319442940, −6.21532568445604949993832185577, −5.92365999219625160161204773627, −5.68406590572197503629335054721, −5.01777702992843472652753311492, −3.95362846861699277713048297047, −3.32499173102226489993322310894, −2.60732321400380514698232162966, −2.54814639524729726420235275367, −1.91499391943724190431809738986, −1.31299646496539755696952707505,
1.31299646496539755696952707505, 1.91499391943724190431809738986, 2.54814639524729726420235275367, 2.60732321400380514698232162966, 3.32499173102226489993322310894, 3.95362846861699277713048297047, 5.01777702992843472652753311492, 5.68406590572197503629335054721, 5.92365999219625160161204773627, 6.21532568445604949993832185577, 7.07219297765751453787319442940, 7.09393507978738512689455294188, 7.84105277319775683298564600351, 8.598070929724526489489679568355, 8.699385849675814534459482604763, 9.188914032787288603776484306400, 9.324793331059242290795139831263, 9.842038728476668612465377262783, 10.16951030776102695258004177625, 10.27962439364336452354976674814
