| L(s) = 1 | − 4·3-s + 4·7-s + 6·9-s − 8·13-s − 16-s + 4·19-s − 16·21-s + 4·27-s − 20·31-s + 4·37-s + 32·39-s + 4·48-s + 8·49-s − 16·57-s + 32·61-s + 24·63-s − 20·67-s + 4·73-s − 40·79-s − 37·81-s − 32·91-s + 80·93-s + 28·97-s + 4·109-s − 16·111-s − 4·112-s − 48·117-s + ⋯ |
| L(s) = 1 | − 2.30·3-s + 1.51·7-s + 2·9-s − 2.21·13-s − 1/4·16-s + 0.917·19-s − 3.49·21-s + 0.769·27-s − 3.59·31-s + 0.657·37-s + 5.12·39-s + 0.577·48-s + 8/7·49-s − 2.11·57-s + 4.09·61-s + 3.02·63-s − 2.44·67-s + 0.468·73-s − 4.50·79-s − 4.11·81-s − 3.35·91-s + 8.29·93-s + 2.84·97-s + 0.383·109-s − 1.51·111-s − 0.377·112-s − 4.43·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2313441 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2313441 ^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2065589846\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2065589846\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| good | 2 | $C_2^3$ | \( 1 + T^{4} + p^{4} T^{8} \) |
| 5 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 8 T^{2} + p^{2} T^{4} )( 1 + 8 T^{2} + p^{2} T^{4} ) \) |
| 7 | $C_2^2$ | \( ( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^3$ | \( 1 - 206 T^{4} + p^{4} T^{8} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^3$ | \( 1 + 2722 T^{4} + p^{4} T^{8} \) |
| 43 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 + 1666 T^{4} + p^{4} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 - 74 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^3$ | \( 1 + 3442 T^{4} + p^{4} T^{8} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^3$ | \( 1 + 5794 T^{4} + p^{4} T^{8} \) |
| 73 | $C_2^2$ | \( ( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{4} \) |
| 83 | $C_2^3$ | \( 1 - 3374 T^{4} + p^{4} T^{8} \) |
| 89 | $C_2^3$ | \( 1 - 15518 T^{4} + p^{4} T^{8} \) |
| 97 | $C_2^2$ | \( ( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−12.04534595841437143990557330796, −11.77109277835793986615228769704, −11.63744982230134169591864503918, −11.30675612783787748154691379910, −11.27655515850053202120672354428, −10.85600526130536181306413773038, −10.33856833734465824807928151464, −10.08889150269282229426962884314, −10.02155482040305598488882210727, −9.211180387486936601493784758346, −9.046220776353028334574936459101, −8.664988877066061889315090811529, −8.061664789178817862340467456829, −7.64074234927874492951923188733, −7.27446410265545340213486625433, −7.09147578880902535888251385419, −6.70588095140817211226667910466, −5.91663455141999524000695565236, −5.60369323473452934674015467315, −5.27301574152984749826579822970, −5.19091605701394411623288195776, −4.53048237557237169882638974755, −4.20924296412302041216682511933, −3.05896449183021981084934127189, −2.01645263655581689352851342192,
2.01645263655581689352851342192, 3.05896449183021981084934127189, 4.20924296412302041216682511933, 4.53048237557237169882638974755, 5.19091605701394411623288195776, 5.27301574152984749826579822970, 5.60369323473452934674015467315, 5.91663455141999524000695565236, 6.70588095140817211226667910466, 7.09147578880902535888251385419, 7.27446410265545340213486625433, 7.64074234927874492951923188733, 8.061664789178817862340467456829, 8.664988877066061889315090811529, 9.046220776353028334574936459101, 9.211180387486936601493784758346, 10.02155482040305598488882210727, 10.08889150269282229426962884314, 10.33856833734465824807928151464, 10.85600526130536181306413773038, 11.27655515850053202120672354428, 11.30675612783787748154691379910, 11.63744982230134169591864503918, 11.77109277835793986615228769704, 12.04534595841437143990557330796
