| L(s) = 1 | − 2·3-s − 4·5-s − 6·7-s + 2·9-s − 4·11-s − 6·13-s + 8·15-s + 2·17-s + 12·21-s − 2·23-s + 11·25-s − 6·27-s + 8·33-s + 24·35-s + 2·37-s + 12·39-s + 20·41-s + 10·43-s − 8·45-s + 6·47-s + 18·49-s − 4·51-s − 10·53-s + 16·55-s − 12·63-s + 24·65-s − 2·67-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 1.78·5-s − 2.26·7-s + 2/3·9-s − 1.20·11-s − 1.66·13-s + 2.06·15-s + 0.485·17-s + 2.61·21-s − 0.417·23-s + 11/5·25-s − 1.15·27-s + 1.39·33-s + 4.05·35-s + 0.328·37-s + 1.92·39-s + 3.12·41-s + 1.52·43-s − 1.19·45-s + 0.875·47-s + 18/7·49-s − 0.560·51-s − 1.37·53-s + 2.15·55-s − 1.51·63-s + 2.97·65-s − 0.244·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 138 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
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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
−9.725339179640002236282433632100, −9.370759991062562532477992399429, −8.659798887324668277105140805649, −7.87122297478285388268026281088, −7.85476255025311309057962383350, −7.35014287573872070486759221161, −7.06455078501503521502100936625, −6.71197145528247694689250592006, −5.90420232710634784222438576496, −5.82342662629659944558900974761, −5.43451323791738122607526773711, −4.57781884727067379168125543282, −4.36093616263909165945957349726, −3.91496179884990342002437619200, −3.25049147325596216711198983869, −2.74361901804286879867873633255, −2.50626595260318498511668047155, −0.878529256752267838504441193761, 0, 0,
0.878529256752267838504441193761, 2.50626595260318498511668047155, 2.74361901804286879867873633255, 3.25049147325596216711198983869, 3.91496179884990342002437619200, 4.36093616263909165945957349726, 4.57781884727067379168125543282, 5.43451323791738122607526773711, 5.82342662629659944558900974761, 5.90420232710634784222438576496, 6.71197145528247694689250592006, 7.06455078501503521502100936625, 7.35014287573872070486759221161, 7.85476255025311309057962383350, 7.87122297478285388268026281088, 8.659798887324668277105140805649, 9.370759991062562532477992399429, 9.725339179640002236282433632100
