| L(s) = 1 | − 3·5-s − 7-s − 3·9-s + 3·11-s + 13-s + 12·17-s + 8·19-s − 3·23-s + 5·25-s − 3·29-s + 5·31-s + 3·35-s + 4·37-s − 3·41-s − 43-s + 9·45-s − 9·47-s + 7·49-s − 12·53-s − 9·55-s − 3·59-s + 13·61-s + 3·63-s − 3·65-s − 7·67-s + 24·71-s − 20·73-s + ⋯ |
| L(s) = 1 | − 1.34·5-s − 0.377·7-s − 9-s + 0.904·11-s + 0.277·13-s + 2.91·17-s + 1.83·19-s − 0.625·23-s + 25-s − 0.557·29-s + 0.898·31-s + 0.507·35-s + 0.657·37-s − 0.468·41-s − 0.152·43-s + 1.34·45-s − 1.31·47-s + 49-s − 1.64·53-s − 1.21·55-s − 0.390·59-s + 1.66·61-s + 0.377·63-s − 0.372·65-s − 0.855·67-s + 2.84·71-s − 2.34·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20736 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20736 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9327316913\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9327316913\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - T - 12 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 3 T - 14 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 3 T - 20 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 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 | $C_2^2$ | \( 1 + 3 T - 32 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + T - 42 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 9 T + 34 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 3 T - 50 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 7 T - 18 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 11 T + 42 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 9 T - 2 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 11 T + 24 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
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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
−13.63939936948839066671965589658, −12.49753018174528950969373641326, −12.17938940567116157236683203657, −11.98128141849072251647111542140, −11.35052065175557841697075784163, −11.17521245136674454721478943905, −10.17307402423722297079296465044, −9.686296243206800738249221035342, −9.419095640760460642674684366510, −8.451398717742178608733058238491, −8.037768318414551450244332052876, −7.71215399010766595203941265039, −7.06442974047196561827728526342, −6.27944521460285198715481266591, −5.62145106335824226285795652209, −5.13111030909990043887555761680, −4.04531457728349794313523609829, −3.34725078321806087360246920646, −3.12456873246481322824104634200, −1.10309822942172362190392132849,
1.10309822942172362190392132849, 3.12456873246481322824104634200, 3.34725078321806087360246920646, 4.04531457728349794313523609829, 5.13111030909990043887555761680, 5.62145106335824226285795652209, 6.27944521460285198715481266591, 7.06442974047196561827728526342, 7.71215399010766595203941265039, 8.037768318414551450244332052876, 8.451398717742178608733058238491, 9.419095640760460642674684366510, 9.686296243206800738249221035342, 10.17307402423722297079296465044, 11.17521245136674454721478943905, 11.35052065175557841697075784163, 11.98128141849072251647111542140, 12.17938940567116157236683203657, 12.49753018174528950969373641326, 13.63939936948839066671965589658
