| L(s) = 1 | − 2-s − 3-s + 2·5-s + 6-s − 7-s + 8-s − 2·10-s + 14-s − 2·15-s − 16-s + 21-s + 4·23-s − 24-s + 3·25-s + 27-s + 29-s + 2·30-s − 2·35-s + 2·40-s + 41-s − 42-s + 43-s − 4·46-s + 47-s + 48-s − 3·50-s − 54-s + ⋯ |
| L(s) = 1 | − 2-s − 3-s + 2·5-s + 6-s − 7-s + 8-s − 2·10-s + 14-s − 2·15-s − 16-s + 21-s + 4·23-s − 24-s + 3·25-s + 27-s + 29-s + 2·30-s − 2·35-s + 2·40-s + 41-s − 42-s + 43-s − 4·46-s + 47-s + 48-s − 3·50-s − 54-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1587600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1587600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5960001155\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5960001155\) |
| \(L(1)\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 23 | $C_1$ | \( ( 1 - T )^{4} \) |
| 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 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 41 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 47 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
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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
−9.934233195470402453857241428281, −9.676677298453753994436027749088, −9.219149705161012387304631671614, −9.079205349113882384530626256772, −8.758741674222259445643648626974, −8.335134978115309031605779819722, −7.33477807684882512473919782155, −7.27880299109105787386317056971, −6.70395650358065671509230206101, −6.45779692556955817692219893279, −5.92380666807666137504429657048, −5.59723659241164069648151583664, −5.12696519920336464650199977840, −4.72467365931598154474555656786, −4.36485917851616584587997433626, −3.05709666582319265364724981590, −2.98729812867865998246408079009, −2.37082301713530552093769824961, −1.19248004057010032279897048169, −1.09837548045772675850030893145,
1.09837548045772675850030893145, 1.19248004057010032279897048169, 2.37082301713530552093769824961, 2.98729812867865998246408079009, 3.05709666582319265364724981590, 4.36485917851616584587997433626, 4.72467365931598154474555656786, 5.12696519920336464650199977840, 5.59723659241164069648151583664, 5.92380666807666137504429657048, 6.45779692556955817692219893279, 6.70395650358065671509230206101, 7.27880299109105787386317056971, 7.33477807684882512473919782155, 8.335134978115309031605779819722, 8.758741674222259445643648626974, 9.079205349113882384530626256772, 9.219149705161012387304631671614, 9.676677298453753994436027749088, 9.934233195470402453857241428281
