| L(s) = 1 | + 2-s + 4-s − 3·5-s − 2·7-s + 8-s + 9-s − 3·10-s − 2·14-s + 16-s + 18-s + 2·19-s − 3·20-s − 25-s − 2·28-s + 32-s + 6·35-s + 36-s − 8·37-s + 2·38-s − 3·40-s − 10·43-s − 3·45-s − 3·49-s − 50-s + 18·53-s − 2·56-s − 2·63-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 1.34·5-s − 0.755·7-s + 0.353·8-s + 1/3·9-s − 0.948·10-s − 0.534·14-s + 1/4·16-s + 0.235·18-s + 0.458·19-s − 0.670·20-s − 1/5·25-s − 0.377·28-s + 0.176·32-s + 1.01·35-s + 1/6·36-s − 1.31·37-s + 0.324·38-s − 0.474·40-s − 1.52·43-s − 0.447·45-s − 3/7·49-s − 0.141·50-s + 2.47·53-s − 0.267·56-s − 0.251·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189728 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189728 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 7 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 11 | $C_2$ | \( 1 + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 28 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 19 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 119 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 100 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 11 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + 8 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
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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
−8.691138987977261454872612422238, −8.439922803308813015646259564729, −7.78192261883320332308639784013, −7.40043628747859289771334596961, −6.85763174552959048181386789668, −6.68328742945276245081036246359, −5.72866037368370315951456694514, −5.50745823427368181481588864498, −4.69166193804206927533777331219, −4.19156990706852376440299622322, −3.66980787206907248985672520920, −3.29630460391738594172950738599, −2.54076353282660633066138649522, −1.48401118787311787717214151038, 0,
1.48401118787311787717214151038, 2.54076353282660633066138649522, 3.29630460391738594172950738599, 3.66980787206907248985672520920, 4.19156990706852376440299622322, 4.69166193804206927533777331219, 5.50745823427368181481588864498, 5.72866037368370315951456694514, 6.68328742945276245081036246359, 6.85763174552959048181386789668, 7.40043628747859289771334596961, 7.78192261883320332308639784013, 8.439922803308813015646259564729, 8.691138987977261454872612422238
