| L(s) = 1 | − 2-s + 3·5-s + 7-s + 8-s − 3·10-s − 3·11-s + 4·13-s − 14-s − 16-s + 4·19-s + 3·22-s − 6·23-s + 5·25-s − 4·26-s + 6·29-s − 5·31-s + 3·35-s + 4·37-s − 4·38-s + 3·40-s − 6·41-s + 10·43-s + 6·46-s + 6·47-s + 7·49-s − 5·50-s − 18·53-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.34·5-s + 0.377·7-s + 0.353·8-s − 0.948·10-s − 0.904·11-s + 1.10·13-s − 0.267·14-s − 1/4·16-s + 0.917·19-s + 0.639·22-s − 1.25·23-s + 25-s − 0.784·26-s + 1.11·29-s − 0.898·31-s + 0.507·35-s + 0.657·37-s − 0.648·38-s + 0.474·40-s − 0.937·41-s + 1.52·43-s + 0.884·46-s + 0.875·47-s + 49-s − 0.707·50-s − 2.47·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26244 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26244 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.061975282\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.061975282\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 4 T + 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 6 T + 7 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 5 T - 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 6 T - 5 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 10 T + 57 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 6 T - 11 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 12 T + 85 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 8 T + 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 14 T + 129 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 3 T - 74 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - T - 96 T^{2} - p T^{3} + p^{2} T^{4} \) |
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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
−13.51703282903545428007093410731, −12.75178558925765669159719781894, −12.00455734165583297223534542533, −11.63360400216876953770638969512, −10.72647343618202831450071557671, −10.61363673681156605231795676965, −10.12893922575711505368266899299, −9.549727085724035416741753721477, −9.051656500452837115222561683600, −8.684033332862363744896073032803, −7.86696656695305832468540019692, −7.67003970350535586772668150124, −6.79669319985080365079294648711, −5.88500719907871205888044515656, −5.87094293569908246181183890819, −4.98561863570386518508403707012, −4.26276781331185966554912181940, −3.18664569842764989663889374755, −2.26057032192534261335558596218, −1.32677919860022877648496519353,
1.32677919860022877648496519353, 2.26057032192534261335558596218, 3.18664569842764989663889374755, 4.26276781331185966554912181940, 4.98561863570386518508403707012, 5.87094293569908246181183890819, 5.88500719907871205888044515656, 6.79669319985080365079294648711, 7.67003970350535586772668150124, 7.86696656695305832468540019692, 8.684033332862363744896073032803, 9.051656500452837115222561683600, 9.549727085724035416741753721477, 10.12893922575711505368266899299, 10.61363673681156605231795676965, 10.72647343618202831450071557671, 11.63360400216876953770638969512, 12.00455734165583297223534542533, 12.75178558925765669159719781894, 13.51703282903545428007093410731
