| L(s) = 1 | − 2·5-s + 4·11-s − 12·19-s − 25-s + 12·29-s + 4·31-s − 4·41-s − 49-s − 8·55-s + 16·59-s − 20·61-s − 12·71-s − 24·79-s − 20·89-s + 24·95-s − 20·101-s + 28·109-s − 10·121-s + 12·125-s + 127-s + 131-s + 137-s + 139-s − 24·145-s + 149-s + 151-s − 8·155-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 1.20·11-s − 2.75·19-s − 1/5·25-s + 2.22·29-s + 0.718·31-s − 0.624·41-s − 1/7·49-s − 1.07·55-s + 2.08·59-s − 2.56·61-s − 1.42·71-s − 2.70·79-s − 2.11·89-s + 2.46·95-s − 1.99·101-s + 2.68·109-s − 0.909·121-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 − 0.642·155-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25401600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25401600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4956298161\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4956298161\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| good | 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 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 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
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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
−8.364793209251106933293610444318, −8.207059025462696215488154000710, −7.84557151625693392196762749095, −7.13718687999467398681961227892, −7.02850262169755123748291515589, −6.61618962127318062346026291891, −6.37978254516918660030836575865, −5.85916611282324796695007356060, −5.77624094091176599376352957521, −4.86791495551848823663244716611, −4.55586544352857365544577275102, −4.40102392997775245018299686935, −3.99615050969032645169286636816, −3.67504785331634815609148757602, −3.05759610798188764377708571982, −2.64800743778501957148776515894, −2.24609220186170019907254664753, −1.45628896097583109959931681246, −1.22128118614708430747711197718, −0.19784682898305887768992090193,
0.19784682898305887768992090193, 1.22128118614708430747711197718, 1.45628896097583109959931681246, 2.24609220186170019907254664753, 2.64800743778501957148776515894, 3.05759610798188764377708571982, 3.67504785331634815609148757602, 3.99615050969032645169286636816, 4.40102392997775245018299686935, 4.55586544352857365544577275102, 4.86791495551848823663244716611, 5.77624094091176599376352957521, 5.85916611282324796695007356060, 6.37978254516918660030836575865, 6.61618962127318062346026291891, 7.02850262169755123748291515589, 7.13718687999467398681961227892, 7.84557151625693392196762749095, 8.207059025462696215488154000710, 8.364793209251106933293610444318
