| L(s) = 1 | + 3·3-s + 6·5-s − 6·7-s + 6·9-s + 3·11-s + 4·13-s + 18·15-s − 18·21-s + 19·25-s + 9·27-s + 6·29-s + 9·33-s − 36·35-s − 4·37-s + 12·39-s − 9·41-s − 9·43-s + 36·45-s − 12·47-s + 17·49-s + 18·55-s + 15·59-s + 8·61-s − 36·63-s + 24·65-s + 15·67-s − 12·71-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 2.68·5-s − 2.26·7-s + 2·9-s + 0.904·11-s + 1.10·13-s + 4.64·15-s − 3.92·21-s + 19/5·25-s + 1.73·27-s + 1.11·29-s + 1.56·33-s − 6.08·35-s − 0.657·37-s + 1.92·39-s − 1.40·41-s − 1.37·43-s + 5.36·45-s − 1.75·47-s + 17/7·49-s + 2.42·55-s + 1.95·59-s + 1.02·61-s − 4.53·63-s + 2.97·65-s + 1.83·67-s − 1.42·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 331776 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 331776 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.029469549\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.029469549\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 6 T + 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 4 T + 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 31 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 35 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 6 T + 41 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 9 T + 68 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 9 T + 70 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 12 T + 97 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 15 T + 166 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 8 T + 3 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 15 T + 142 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 6 T + 91 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 12 T + 61 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 13 T + 72 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
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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
−10.31745065572463801169795924465, −10.14069559994496422658207913127, −10.04374173775015300804833092415, −9.686960115219519920153797334249, −9.078878130059815791317740130140, −9.076504975622358508676270055850, −8.463291896966235723150019959319, −8.219460303645119564646346064531, −6.96214460241739106284324130843, −6.87851483358594982965262773710, −6.42535037141320614319280356163, −6.20790185086092601676890831339, −5.53199019256473956643755450123, −5.02609093084749221952817661269, −3.93662465711349724506420647441, −3.62363196865782396133594269974, −2.87560496125806906661118029770, −2.76269522057821848468888476798, −1.77708389283447358557893073027, −1.42613130742504914712939237712,
1.42613130742504914712939237712, 1.77708389283447358557893073027, 2.76269522057821848468888476798, 2.87560496125806906661118029770, 3.62363196865782396133594269974, 3.93662465711349724506420647441, 5.02609093084749221952817661269, 5.53199019256473956643755450123, 6.20790185086092601676890831339, 6.42535037141320614319280356163, 6.87851483358594982965262773710, 6.96214460241739106284324130843, 8.219460303645119564646346064531, 8.463291896966235723150019959319, 9.076504975622358508676270055850, 9.078878130059815791317740130140, 9.686960115219519920153797334249, 10.04374173775015300804833092415, 10.14069559994496422658207913127, 10.31745065572463801169795924465
