| L(s) = 1 | + 2·2-s + 2·4-s − 3·5-s + 4·8-s + 3·9-s − 6·10-s + 6·11-s + 2·13-s + 8·16-s + 4·17-s + 6·18-s + 5·19-s − 6·20-s + 12·22-s − 3·23-s + 5·25-s + 4·26-s − 10·29-s − 3·31-s + 8·32-s + 8·34-s + 6·36-s + 4·37-s + 10·38-s − 12·40-s + 12·41-s − 2·43-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s − 1.34·5-s + 1.41·8-s + 9-s − 1.89·10-s + 1.80·11-s + 0.554·13-s + 2·16-s + 0.970·17-s + 1.41·18-s + 1.14·19-s − 1.34·20-s + 2.55·22-s − 0.625·23-s + 25-s + 0.784·26-s − 1.85·29-s − 0.538·31-s + 1.41·32-s + 1.37·34-s + 36-s + 0.657·37-s + 1.62·38-s − 1.89·40-s + 1.87·41-s − 0.304·43-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
| \(L(1)\) |
\(\approx\) |
\(4.776777761\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.776777761\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - p T + p T^{2} - p^{2} T^{3} + p^{2} T^{4} \) |
| 3 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $C_2^2$ | \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 6 T + 25 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 4 T - T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 5 T + 6 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 3 T - 14 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 3 T - 22 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 4 T - 21 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 7 T + 2 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 9 T + 28 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 8 T + 5 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 10 T + 39 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 6 T - 31 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 13 T + 96 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 3 T - 70 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 15 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 3 T - 80 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
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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
−10.85136710588636109608076253036, −10.65692092460855196745324440424, −9.913449355444814881784477685814, −9.568364925883983481630577943600, −9.167391887771824174634402017319, −8.470982317285759574967752985939, −8.032324588200659913296266724006, −7.35196007869106240365006618850, −7.18908742508590601313968260101, −7.15366505102637705856941077257, −5.99197996786868933247458472617, −5.81999417995825630018858231590, −5.31296843448578688893409175778, −4.39517148428385295574554083289, −4.20341065449296861761035057529, −3.90337222200630746688036814772, −3.57050520560488258897162781027, −2.80261427554416638192209594583, −1.50938155335232479980636395902, −1.20919581381145784545195474837,
1.20919581381145784545195474837, 1.50938155335232479980636395902, 2.80261427554416638192209594583, 3.57050520560488258897162781027, 3.90337222200630746688036814772, 4.20341065449296861761035057529, 4.39517148428385295574554083289, 5.31296843448578688893409175778, 5.81999417995825630018858231590, 5.99197996786868933247458472617, 7.15366505102637705856941077257, 7.18908742508590601313968260101, 7.35196007869106240365006618850, 8.032324588200659913296266724006, 8.470982317285759574967752985939, 9.167391887771824174634402017319, 9.568364925883983481630577943600, 9.913449355444814881784477685814, 10.65692092460855196745324440424, 10.85136710588636109608076253036
