| L(s) = 1 | + 2-s + 3-s − 5-s + 6-s − 7-s − 8-s − 10-s − 14-s − 15-s − 16-s − 21-s − 23-s − 24-s − 27-s + 29-s − 30-s + 35-s + 40-s + 41-s − 42-s + 2·43-s − 46-s − 47-s − 48-s + 49-s − 54-s + 56-s + ⋯ |
| L(s) = 1 | + 2-s + 3-s − 5-s + 6-s − 7-s − 8-s − 10-s − 14-s − 15-s − 16-s − 21-s − 23-s − 24-s − 27-s + 29-s − 30-s + 35-s + 40-s + 41-s − 42-s + 2·43-s − 46-s − 47-s − 48-s + 49-s − 54-s + 56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6674777657\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6674777657\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 3 | $C_2$ | \( 1 - T + T^{2} \) |
| 5 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 7 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 23 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 29 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 41 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 47 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 83 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.25266003726459056934228925380, −12.63618427465138715563032354413, −12.34233518978514723443508067396, −11.93483636824218202361140453035, −11.32059066798460248258732799065, −10.87563652432827529726951676893, −9.877856881384228442551797346298, −9.755306957471470115556646605831, −9.052683058898347171661999093782, −8.548958346639384954526037770529, −8.175583683876575953362057897275, −7.48473382694494253624140028139, −6.99564495359185224065559830264, −5.96797693585416330644713210577, −5.96541110648486450872095668393, −4.86178503697550767939988741389, −4.02505199161513633549089042307, −3.87921310680456161343166166782, −3.01554335708622184459064059850, −2.52796415098278775375283093093,
2.52796415098278775375283093093, 3.01554335708622184459064059850, 3.87921310680456161343166166782, 4.02505199161513633549089042307, 4.86178503697550767939988741389, 5.96541110648486450872095668393, 5.96797693585416330644713210577, 6.99564495359185224065559830264, 7.48473382694494253624140028139, 8.175583683876575953362057897275, 8.548958346639384954526037770529, 9.052683058898347171661999093782, 9.755306957471470115556646605831, 9.877856881384228442551797346298, 10.87563652432827529726951676893, 11.32059066798460248258732799065, 11.93483636824218202361140453035, 12.34233518978514723443508067396, 12.63618427465138715563032354413, 13.25266003726459056934228925380
