| L(s) = 1 | + 2·5-s − 9-s + 16·19-s − 25-s + 12·29-s − 16·31-s + 12·41-s − 2·45-s − 2·49-s − 12·61-s − 32·71-s + 16·79-s + 81-s + 20·89-s + 32·95-s + 28·101-s + 20·109-s − 22·121-s − 12·125-s + 127-s + 131-s + 137-s + 139-s + 24·145-s + 149-s + 151-s − 32·155-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 1/3·9-s + 3.67·19-s − 1/5·25-s + 2.22·29-s − 2.87·31-s + 1.87·41-s − 0.298·45-s − 2/7·49-s − 1.53·61-s − 3.79·71-s + 1.80·79-s + 1/9·81-s + 2.11·89-s + 3.28·95-s + 2.78·101-s + 1.91·109-s − 2·121-s − 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1.99·145-s + 0.0819·149-s + 0.0813·151-s − 2.57·155-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.210334227\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.210334227\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−11.41743970778774975400303966308, −10.57755576727606547849543533034, −10.31691619638260937462241075078, −9.929820598187428319125980332286, −9.251093571579624393858827862030, −9.121682277424501562838310149212, −8.861824877114813117932923518761, −7.69100625337343659531140064385, −7.58046593760750272291590356017, −7.42734279973460185432009308593, −6.33194324103217618765407995558, −6.19522811229739267154054994597, −5.40168999533577827146192711099, −5.29939454441125981195058243999, −4.66032129314467812190984027118, −3.76388813856350738805375389722, −3.15891568248029124408836802439, −2.74322786618082339973145301394, −1.75732140595578105415486323736, −0.999355934092705003166772031466,
0.999355934092705003166772031466, 1.75732140595578105415486323736, 2.74322786618082339973145301394, 3.15891568248029124408836802439, 3.76388813856350738805375389722, 4.66032129314467812190984027118, 5.29939454441125981195058243999, 5.40168999533577827146192711099, 6.19522811229739267154054994597, 6.33194324103217618765407995558, 7.42734279973460185432009308593, 7.58046593760750272291590356017, 7.69100625337343659531140064385, 8.861824877114813117932923518761, 9.121682277424501562838310149212, 9.251093571579624393858827862030, 9.929820598187428319125980332286, 10.31691619638260937462241075078, 10.57755576727606547849543533034, 11.41743970778774975400303966308
