| L(s) = 1 | − 2·2-s + 2·4-s − 2·5-s − 6·7-s + 4·10-s + 2·11-s − 4·13-s + 12·14-s − 4·16-s − 2·17-s + 6·19-s − 4·20-s − 4·22-s + 2·23-s − 25-s + 8·26-s − 12·28-s − 14·29-s + 8·32-s + 4·34-s + 12·35-s − 12·37-s − 12·38-s + 8·43-s + 4·44-s − 4·46-s − 14·47-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s − 0.894·5-s − 2.26·7-s + 1.26·10-s + 0.603·11-s − 1.10·13-s + 3.20·14-s − 16-s − 0.485·17-s + 1.37·19-s − 0.894·20-s − 0.852·22-s + 0.417·23-s − 1/5·25-s + 1.56·26-s − 2.26·28-s − 2.59·29-s + 1.41·32-s + 0.685·34-s + 2.02·35-s − 1.97·37-s − 1.94·38-s + 1.21·43-s + 0.603·44-s − 0.589·46-s − 2.04·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 518400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 518400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 66 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 162 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 22 T + 242 T^{2} + 22 p T^{3} + p^{2} T^{4} \) |
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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
−9.860292456320144955217588739372, −9.674422496231680280720683627009, −9.359720314011369485442164279945, −9.236980611872958340383987036003, −8.497223744181190731333463216608, −8.157539091289481418000010203263, −7.42349037622531566739298720648, −7.28601148568030039277684638473, −6.77742169733307048759500197434, −6.71944664485963193718107545436, −5.65151562715689229601076005751, −5.53526374764812640783799225316, −4.59012254762310538829505268378, −3.97897270449761328231240330488, −3.48070920141400045756282866034, −3.07223970554371371951409765966, −2.29034501122603879683994059558, −1.39071112323543221956068391186, 0, 0,
1.39071112323543221956068391186, 2.29034501122603879683994059558, 3.07223970554371371951409765966, 3.48070920141400045756282866034, 3.97897270449761328231240330488, 4.59012254762310538829505268378, 5.53526374764812640783799225316, 5.65151562715689229601076005751, 6.71944664485963193718107545436, 6.77742169733307048759500197434, 7.28601148568030039277684638473, 7.42349037622531566739298720648, 8.157539091289481418000010203263, 8.497223744181190731333463216608, 9.236980611872958340383987036003, 9.359720314011369485442164279945, 9.674422496231680280720683627009, 9.860292456320144955217588739372
