| L(s) = 1 | + 3·4-s − 2·11-s + 5·16-s + 12·19-s + 12·29-s + 8·31-s − 20·41-s − 6·44-s + 10·49-s − 8·59-s − 12·61-s + 3·64-s + 36·76-s + 20·79-s − 28·101-s + 4·109-s + 36·116-s + 3·121-s + 24·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | + 3/2·4-s − 0.603·11-s + 5/4·16-s + 2.75·19-s + 2.22·29-s + 1.43·31-s − 3.12·41-s − 0.904·44-s + 10/7·49-s − 1.04·59-s − 1.53·61-s + 3/8·64-s + 4.12·76-s + 2.25·79-s − 2.78·101-s + 0.383·109-s + 3.34·116-s + 3/11·121-s + 2.15·124-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6125625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6125625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.467104323\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.467104323\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 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
−9.064772618816293519184799270513, −8.757033627962422923244267847932, −8.085054617155687207817875365553, −7.86460325274192278819449964194, −7.77650150645533024114670686994, −7.05144308678205909703116615342, −6.81458709377288916645525663819, −6.58400181546080387468159689340, −6.14786168815816536174015883674, −5.52324681384973635070138776531, −5.33300021623181321636655425015, −4.84623218618087504689168216774, −4.47725309699010482868661396691, −3.66254637291172556164165333417, −3.19536578112855500461539776202, −2.82947706574943125181324552895, −2.68241370116663028907022659859, −1.80067723740961108311090070147, −1.35300550056206041348954484465, −0.72821514763032145207897256027,
0.72821514763032145207897256027, 1.35300550056206041348954484465, 1.80067723740961108311090070147, 2.68241370116663028907022659859, 2.82947706574943125181324552895, 3.19536578112855500461539776202, 3.66254637291172556164165333417, 4.47725309699010482868661396691, 4.84623218618087504689168216774, 5.33300021623181321636655425015, 5.52324681384973635070138776531, 6.14786168815816536174015883674, 6.58400181546080387468159689340, 6.81458709377288916645525663819, 7.05144308678205909703116615342, 7.77650150645533024114670686994, 7.86460325274192278819449964194, 8.085054617155687207817875365553, 8.757033627962422923244267847932, 9.064772618816293519184799270513
