| L(s) = 1 | + 2-s − 4-s + 2·5-s − 3·7-s − 3·8-s − 9-s + 2·10-s + 11-s − 5·13-s − 3·14-s − 16-s − 18-s − 19-s − 2·20-s + 22-s − 4·23-s − 6·25-s − 5·26-s + 3·28-s + 29-s + 3·31-s + 5·32-s − 6·35-s + 36-s − 11·37-s − 38-s − 6·40-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s + 0.894·5-s − 1.13·7-s − 1.06·8-s − 1/3·9-s + 0.632·10-s + 0.301·11-s − 1.38·13-s − 0.801·14-s − 1/4·16-s − 0.235·18-s − 0.229·19-s − 0.447·20-s + 0.213·22-s − 0.834·23-s − 6/5·25-s − 0.980·26-s + 0.566·28-s + 0.185·29-s + 0.538·31-s + 0.883·32-s − 1.01·35-s + 1/6·36-s − 1.80·37-s − 0.162·38-s − 0.948·40-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 61632 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 61632 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 - T + p T^{2} \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 107 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 6 T + p T^{2} ) \) |
| good | 5 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) |
| 7 | $D_{4}$ | \( 1 + 3 T + 15 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $D_{4}$ | \( 1 + 5 T + 19 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $D_{4}$ | \( 1 + 4 T + 12 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - T + 45 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 3 T + 37 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 11 T + 91 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 3 T + 31 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + T - 49 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 2 T + 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_4$ | \( 1 - 15 T + 140 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 7 T + 66 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 5 T + 12 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 3 T + 66 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 13 T + 110 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 7 T - 6 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 102 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 3 T - 11 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 8 T + 160 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
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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
−14.6091878594, −14.1014556909, −13.8931874519, −13.4749994381, −13.0832370082, −12.5205723540, −12.1435464460, −11.9404883460, −11.2615142087, −10.3861005084, −10.0732043076, −9.63934093423, −9.41073107611, −8.78617390873, −8.23828118197, −7.58842043754, −6.84155728783, −6.38606150458, −5.90446005158, −5.41059174038, −4.85266821823, −4.09488477837, −3.49886114853, −2.76275831343, −2.01699489171, 0,
2.01699489171, 2.76275831343, 3.49886114853, 4.09488477837, 4.85266821823, 5.41059174038, 5.90446005158, 6.38606150458, 6.84155728783, 7.58842043754, 8.23828118197, 8.78617390873, 9.41073107611, 9.63934093423, 10.0732043076, 10.3861005084, 11.2615142087, 11.9404883460, 12.1435464460, 12.5205723540, 13.0832370082, 13.4749994381, 13.8931874519, 14.1014556909, 14.6091878594
