| L(s) = 1 | − 2·5-s − 8·11-s − 25-s − 12·29-s − 8·31-s + 20·41-s − 2·49-s + 16·55-s − 8·59-s + 4·61-s − 24·79-s − 20·89-s + 4·101-s + 4·109-s + 26·121-s + 12·125-s + 127-s + 131-s + 137-s + 139-s + 24·145-s + 149-s + 151-s + 16·155-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 2.41·11-s − 1/5·25-s − 2.22·29-s − 1.43·31-s + 3.12·41-s − 2/7·49-s + 2.15·55-s − 1.04·59-s + 0.512·61-s − 2.70·79-s − 2.11·89-s + 0.398·101-s + 0.383·109-s + 2.36·121-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 + 1.28·155-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 518400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 518400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5246788188\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5246788188\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 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
−10.80129820806855198684660958484, −10.34732126367003404452157311344, −9.605564628161406731029364131465, −9.482518694768146588210532223282, −8.937430435030389903083480069754, −8.177104534353187535236921364820, −8.119503343433143734311525788350, −7.48504056370313229914684424488, −7.37999553275264887201983989654, −6.96744839759609559170902109064, −5.86061504132680503630264048611, −5.65796269104276392091442659837, −5.47446280495327654978725606812, −4.45734366663742559905138851003, −4.37641700718845026558430007808, −3.56793432555762107040647050628, −3.04865613412047538905424338313, −2.44300734529845398640788981766, −1.79273867690764490936497708859, −0.35992289656517384055802444759,
0.35992289656517384055802444759, 1.79273867690764490936497708859, 2.44300734529845398640788981766, 3.04865613412047538905424338313, 3.56793432555762107040647050628, 4.37641700718845026558430007808, 4.45734366663742559905138851003, 5.47446280495327654978725606812, 5.65796269104276392091442659837, 5.86061504132680503630264048611, 6.96744839759609559170902109064, 7.37999553275264887201983989654, 7.48504056370313229914684424488, 8.119503343433143734311525788350, 8.177104534353187535236921364820, 8.937430435030389903083480069754, 9.482518694768146588210532223282, 9.605564628161406731029364131465, 10.34732126367003404452157311344, 10.80129820806855198684660958484
