| L(s) = 1 | − 2·5-s − 6·13-s + 2·17-s + 5·25-s − 10·29-s + 2·37-s − 20·41-s − 14·49-s − 28·53-s + 10·61-s + 12·65-s + 6·73-s − 4·85-s + 10·89-s + 36·97-s + 4·101-s − 6·109-s − 14·113-s − 22·121-s − 22·125-s + 127-s + 131-s + 137-s + 139-s + 20·145-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 1.66·13-s + 0.485·17-s + 25-s − 1.85·29-s + 0.328·37-s − 3.12·41-s − 2·49-s − 3.84·53-s + 1.28·61-s + 1.48·65-s + 0.702·73-s − 0.433·85-s + 1.05·89-s + 3.65·97-s + 0.398·101-s − 0.574·109-s − 1.31·113-s − 2·121-s − 1.96·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1.66·145-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6718464 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6718464 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 2 T - 13 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 10 T + 71 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 2 T - 33 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 10 T + 39 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 6 T - 37 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 10 T + 11 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{2} \) |
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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
−8.632326023041066561794661252468, −8.199234627009631173491892201221, −7.79515094847959590588506878913, −7.69310086911105794395060229175, −7.28268689162276652758007993137, −6.81117507315177707047311030050, −6.34261767936667054516553443783, −6.25691693908072363262966121520, −5.22724566305105273202413373083, −5.12778135442031525818356400688, −4.89762079980259759784033602099, −4.40731150190944285276890829366, −3.56203580692724898330422010749, −3.56172121734958151694176277093, −3.06769458728378563523590662439, −2.38270260441747612385347589787, −1.85179095146423760832242357565, −1.29162222912781718445665208888, 0, 0,
1.29162222912781718445665208888, 1.85179095146423760832242357565, 2.38270260441747612385347589787, 3.06769458728378563523590662439, 3.56172121734958151694176277093, 3.56203580692724898330422010749, 4.40731150190944285276890829366, 4.89762079980259759784033602099, 5.12778135442031525818356400688, 5.22724566305105273202413373083, 6.25691693908072363262966121520, 6.34261767936667054516553443783, 6.81117507315177707047311030050, 7.28268689162276652758007993137, 7.69310086911105794395060229175, 7.79515094847959590588506878913, 8.199234627009631173491892201221, 8.632326023041066561794661252468
