| L(s) = 1 | − 4·2-s + 8·4-s + 4·7-s − 16·8-s − 4·11-s − 16·14-s + 36·16-s + 16·22-s + 8·23-s + 32·28-s − 64·32-s − 24·37-s − 12·43-s − 32·44-s − 32·46-s + 8·49-s + 4·53-s − 64·56-s + 96·64-s − 4·67-s − 24·71-s + 96·74-s − 16·77-s + 17·81-s + 48·86-s + 64·88-s + 64·92-s + ⋯ |
| L(s) = 1 | − 2.82·2-s + 4·4-s + 1.51·7-s − 5.65·8-s − 1.20·11-s − 4.27·14-s + 9·16-s + 3.41·22-s + 1.66·23-s + 6.04·28-s − 11.3·32-s − 3.94·37-s − 1.82·43-s − 4.82·44-s − 4.71·46-s + 8/7·49-s + 0.549·53-s − 8.55·56-s + 12·64-s − 0.488·67-s − 2.84·71-s + 11.1·74-s − 1.82·77-s + 17/9·81-s + 5.17·86-s + 6.82·88-s + 6.67·92-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1500625 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1500625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.08940827819\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.08940827819\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| good | 2 | $C_2^2$ | \( ( 1 + p T + p T^{2} + p^{2} T^{3} + p^{2} T^{4} )^{2} \) |
| 3 | $C_2^3$ | \( 1 - 17 T^{4} + p^{4} T^{8} \) |
| 11 | $C_2$ | \( ( 1 + T + p T^{2} )^{4} \) |
| 13 | $C_2^3$ | \( 1 + 103 T^{4} + p^{4} T^{8} \) |
| 17 | $C_2^3$ | \( 1 + 263 T^{4} + p^{4} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 + 28 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 49 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 52 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + 12 T + 72 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + 8 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 - 2017 T^{4} + p^{4} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 28 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 + 11 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 + 7538 T^{4} + p^{4} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 + 138 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^3$ | \( 1 + 16903 T^{4} + p^{4} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.32676045744168026200468324578, −11.98832914484567149905474922867, −11.83616758473911985107263360753, −11.45229553346159824489309456196, −11.06694238668489043531651720064, −10.58209303738455575164574949203, −10.50833723239882525662713597838, −10.16812235815551413298297690346, −9.920579216029164867063356025505, −9.238250136225366813809977809960, −9.126247062444299222986819673632, −8.784546719314304664614993136090, −8.524147519473872321912538499874, −8.175085457484967238510962436060, −8.071770588726477605787112555276, −7.45845671624323933270033329055, −6.94729152762131401251943129619, −6.93564282090983517655828207269, −6.04415488762146579249464185117, −5.61138339060652986428905421835, −5.16726999488233885655867866114, −4.76830288810563033065637744208, −3.37951316042159346516949348695, −3.02738172602077541111222636184, −1.81797194049574087688155293048,
1.81797194049574087688155293048, 3.02738172602077541111222636184, 3.37951316042159346516949348695, 4.76830288810563033065637744208, 5.16726999488233885655867866114, 5.61138339060652986428905421835, 6.04415488762146579249464185117, 6.93564282090983517655828207269, 6.94729152762131401251943129619, 7.45845671624323933270033329055, 8.071770588726477605787112555276, 8.175085457484967238510962436060, 8.524147519473872321912538499874, 8.784546719314304664614993136090, 9.126247062444299222986819673632, 9.238250136225366813809977809960, 9.920579216029164867063356025505, 10.16812235815551413298297690346, 10.50833723239882525662713597838, 10.58209303738455575164574949203, 11.06694238668489043531651720064, 11.45229553346159824489309456196, 11.83616758473911985107263360753, 11.98832914484567149905474922867, 12.32676045744168026200468324578
