| L(s) = 1 | − 3·7-s − 2·9-s + 10·11-s + 16·23-s − 9·25-s − 4·29-s + 20·37-s + 2·43-s + 2·49-s − 8·53-s + 6·63-s − 24·67-s + 4·71-s − 30·77-s + 16·79-s − 5·81-s − 20·99-s + 4·107-s − 20·113-s + 53·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + ⋯ |
| L(s) = 1 | − 1.13·7-s − 2/3·9-s + 3.01·11-s + 3.33·23-s − 9/5·25-s − 0.742·29-s + 3.28·37-s + 0.304·43-s + 2/7·49-s − 1.09·53-s + 0.755·63-s − 2.93·67-s + 0.474·71-s − 3.41·77-s + 1.80·79-s − 5/9·81-s − 2.01·99-s + 0.386·107-s − 1.88·113-s + 4.81·121-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 283024 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 283024 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.825864643\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.825864643\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + 3 T + p T^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
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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
−9.084561409834522310688102275122, −8.709339720534257240327886562953, −7.64552271968190118551531459773, −7.62429229152375892340957919668, −6.76474142604661213574836140046, −6.44323514092267885981017350010, −6.21156913138879588632301451179, −5.67409965982210077643645681036, −4.91419379168876950061294361671, −4.15537276955970084853193543262, −3.93593184874987401028815060034, −3.14091634253431942337825536555, −2.82511261070703297705797963037, −1.62016711898696823707962362202, −0.869557064463155631256543449078,
0.869557064463155631256543449078, 1.62016711898696823707962362202, 2.82511261070703297705797963037, 3.14091634253431942337825536555, 3.93593184874987401028815060034, 4.15537276955970084853193543262, 4.91419379168876950061294361671, 5.67409965982210077643645681036, 6.21156913138879588632301451179, 6.44323514092267885981017350010, 6.76474142604661213574836140046, 7.62429229152375892340957919668, 7.64552271968190118551531459773, 8.709339720534257240327886562953, 9.084561409834522310688102275122
