| L(s) = 1 | − 2·4-s + 8·7-s − 6·9-s + 3·16-s − 14·25-s − 16·28-s + 12·36-s − 20·37-s + 4·43-s + 34·49-s − 48·63-s − 4·64-s − 16·67-s − 40·79-s + 27·81-s + 28·100-s − 44·109-s + 24·112-s + 44·121-s + 127-s + 131-s + 137-s + 139-s − 18·144-s + 40·148-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 4-s + 3.02·7-s − 2·9-s + 3/4·16-s − 2.79·25-s − 3.02·28-s + 2·36-s − 3.28·37-s + 0.609·43-s + 34/7·49-s − 6.04·63-s − 1/2·64-s − 1.95·67-s − 4.50·79-s + 3·81-s + 14/5·100-s − 4.21·109-s + 2.26·112-s + 4·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 3/2·144-s + 3.28·148-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7325955753\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7325955753\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| good | 5 | $C_2^2$ | \( ( 1 + 7 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2}( 1 + 7 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 23 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 - 65 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - T + p T^{2} )^{4} \) |
| 47 | $C_2^2$ | \( ( 1 + 19 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 106 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 71 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 98 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + 154 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 166 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 47 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−7.16949163931782490797876191436, −7.07850844113640493192684156298, −6.91344075164067255780572869869, −6.53404776537550041285097144429, −5.91256506288248726423754190319, −5.78645971627265223780756967034, −5.72506712664060025412618134921, −5.66270869606066797162219548838, −5.41047220927664224654783423445, −5.23821865220766797978184886809, −4.70410686785791610149054076297, −4.63962469859591386777720407666, −4.36385820630834318864349436907, −4.26457950349936962349319360799, −4.09279397665848444004050171896, −3.35507946680618720705136800393, −3.29809003246557298704333918087, −3.26446235748117919696487282760, −2.71769643105006620142910272785, −2.18792019972323632835572394809, −1.96982109007826850296924808870, −1.73782153158932564270454481239, −1.57283219389787322956873686084, −0.883621022569072292952080935504, −0.22201275981737702551831725562,
0.22201275981737702551831725562, 0.883621022569072292952080935504, 1.57283219389787322956873686084, 1.73782153158932564270454481239, 1.96982109007826850296924808870, 2.18792019972323632835572394809, 2.71769643105006620142910272785, 3.26446235748117919696487282760, 3.29809003246557298704333918087, 3.35507946680618720705136800393, 4.09279397665848444004050171896, 4.26457950349936962349319360799, 4.36385820630834318864349436907, 4.63962469859591386777720407666, 4.70410686785791610149054076297, 5.23821865220766797978184886809, 5.41047220927664224654783423445, 5.66270869606066797162219548838, 5.72506712664060025412618134921, 5.78645971627265223780756967034, 5.91256506288248726423754190319, 6.53404776537550041285097144429, 6.91344075164067255780572869869, 7.07850844113640493192684156298, 7.16949163931782490797876191436
