| L(s) = 1 | − 9-s + 8·19-s + 12·29-s + 20·41-s − 2·49-s + 12·61-s + 32·79-s + 81-s − 4·89-s − 28·101-s + 20·109-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 10·169-s − 8·171-s + 173-s + 179-s + 181-s + 191-s + ⋯ |
| L(s) = 1 | − 1/3·9-s + 1.83·19-s + 2.22·29-s + 3.12·41-s − 2/7·49-s + 1.53·61-s + 3.60·79-s + 1/9·81-s − 0.423·89-s − 2.78·101-s + 1.91·109-s − 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 0.769·169-s − 0.611·171-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.479682032\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.479682032\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 190 T^{2} + p^{2} T^{4} \) |
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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
−9.760732041280972527856347353013, −9.654292270207471035205419719909, −9.014339424275162161084751947198, −8.980319802170805215763440321635, −8.085502191399673023672854069241, −8.001554278993529444070278222770, −7.63218049132107445008936487280, −7.02120669528881926224263578818, −6.63787881418398969169268982138, −6.23112452798933265342365838147, −5.72723282946656731861682842109, −5.23244228999453151157744291844, −4.95347451618516257564491111813, −4.31653337696327592254722173010, −3.82848213857089380918864401632, −3.28303566753476648187416380439, −2.62126564951844982826337065403, −2.42501775851794867666809742734, −1.23136801108951504994366663802, −0.790843854266940326961438179187,
0.790843854266940326961438179187, 1.23136801108951504994366663802, 2.42501775851794867666809742734, 2.62126564951844982826337065403, 3.28303566753476648187416380439, 3.82848213857089380918864401632, 4.31653337696327592254722173010, 4.95347451618516257564491111813, 5.23244228999453151157744291844, 5.72723282946656731861682842109, 6.23112452798933265342365838147, 6.63787881418398969169268982138, 7.02120669528881926224263578818, 7.63218049132107445008936487280, 8.001554278993529444070278222770, 8.085502191399673023672854069241, 8.980319802170805215763440321635, 9.014339424275162161084751947198, 9.654292270207471035205419719909, 9.760732041280972527856347353013
