| L(s) = 1 | − 5-s + 4·7-s + 6·11-s + 4·13-s + 6·17-s − 14·19-s − 9·23-s + 7·31-s − 4·35-s + 4·37-s + 6·41-s − 2·43-s + 7·49-s − 18·53-s − 6·55-s − 12·59-s + 7·61-s − 4·65-s − 2·67-s − 12·71-s + 4·73-s + 24·77-s + 79-s + 9·83-s − 6·85-s + 12·89-s + 16·91-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.51·7-s + 1.80·11-s + 1.10·13-s + 1.45·17-s − 3.21·19-s − 1.87·23-s + 1.25·31-s − 0.676·35-s + 0.657·37-s + 0.937·41-s − 0.304·43-s + 49-s − 2.47·53-s − 0.809·55-s − 1.56·59-s + 0.896·61-s − 0.496·65-s − 0.244·67-s − 1.42·71-s + 0.468·73-s + 2.73·77-s + 0.112·79-s + 0.987·83-s − 0.650·85-s + 1.27·89-s + 1.67·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2624400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2624400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.891881953\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.891881953\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 6 T + 25 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 4 T + 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 9 T + 58 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 6 T - 5 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 2 T - 39 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 12 T + 85 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 7 T - 12 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 2 T - 63 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - T - 78 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 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 | $C_2^2$ | \( 1 + 8 T - 33 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
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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
−9.599073670647875580890225106251, −8.925267328207665251863436388048, −8.797360495766129746582394319092, −8.382808113225048940027046662983, −7.986123386515366497756441329888, −7.80960608491465966794122101785, −7.39802598763511752249860552070, −6.45904092928053239715152338579, −6.30165999614062061506762570897, −6.20998983155511659215035609453, −5.67840991366308643755309491143, −4.71475621974649754560240220270, −4.58863493562155675783775664763, −4.20470730540135368422914554810, −3.75920651264506611501649039411, −3.37562301298493495889530424173, −2.45650225168527685564653888210, −1.69144646155458567491023220668, −1.63395255269694657377759452445, −0.68854983713981989895663223638,
0.68854983713981989895663223638, 1.63395255269694657377759452445, 1.69144646155458567491023220668, 2.45650225168527685564653888210, 3.37562301298493495889530424173, 3.75920651264506611501649039411, 4.20470730540135368422914554810, 4.58863493562155675783775664763, 4.71475621974649754560240220270, 5.67840991366308643755309491143, 6.20998983155511659215035609453, 6.30165999614062061506762570897, 6.45904092928053239715152338579, 7.39802598763511752249860552070, 7.80960608491465966794122101785, 7.986123386515366497756441329888, 8.382808113225048940027046662983, 8.797360495766129746582394319092, 8.925267328207665251863436388048, 9.599073670647875580890225106251
