L(s) = 1 | − 2-s − 2·3-s + 4-s + 2·5-s + 2·6-s − 2·7-s − 3·8-s + 3·9-s − 2·10-s + 2·11-s − 2·12-s − 6·13-s + 2·14-s − 4·15-s + 16-s − 3·17-s − 3·18-s − 19-s + 2·20-s + 4·21-s − 2·22-s − 7·23-s + 6·24-s + 3·25-s + 6·26-s − 4·27-s − 2·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.15·3-s + 1/2·4-s + 0.894·5-s + 0.816·6-s − 0.755·7-s − 1.06·8-s + 9-s − 0.632·10-s + 0.603·11-s − 0.577·12-s − 1.66·13-s + 0.534·14-s − 1.03·15-s + 1/4·16-s − 0.727·17-s − 0.707·18-s − 0.229·19-s + 0.447·20-s + 0.872·21-s − 0.426·22-s − 1.45·23-s + 1.22·24-s + 3/5·25-s + 1.17·26-s − 0.769·27-s − 0.377·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1334025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1334025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
good | 2 | $D_{4}$ | \( 1 + T + p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 3 T + 32 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + T + 34 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 7 T + 54 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 9 T + 40 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 2 T + 46 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 6 T + 66 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $D_{4}$ | \( 1 - T + 82 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 2 T + 78 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 15 T + 124 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 17 T + 186 T^{2} - 17 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 11 T + 148 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 22 T + 238 T^{2} - 22 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 2 T + 126 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 14 T + 178 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 4 T + 94 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 15 T + 218 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 17 T + 212 T^{2} + 17 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 15 T + 144 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
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
−9.619147841225628573576640357938, −9.513967122264208968458849986444, −8.739180470391774304001035876449, −8.693753344415080044023494634834, −7.86183836477138691924054528738, −7.36721932825129260434292650659, −7.00023991855283618727054549133, −6.62439192388111697068340986083, −6.33124672664491325866903660357, −5.79612347605071384668189312692, −5.55684075551602445238977563965, −5.05129384442759653922738385098, −4.37273487966080637528632387399, −3.96731816426904121676312942968, −3.23581817406617446204227935853, −2.45959341744008594917124935014, −2.12850837763730799097357776973, −1.42753918033326313510965428364, 0, 0,
1.42753918033326313510965428364, 2.12850837763730799097357776973, 2.45959341744008594917124935014, 3.23581817406617446204227935853, 3.96731816426904121676312942968, 4.37273487966080637528632387399, 5.05129384442759653922738385098, 5.55684075551602445238977563965, 5.79612347605071384668189312692, 6.33124672664491325866903660357, 6.62439192388111697068340986083, 7.00023991855283618727054549133, 7.36721932825129260434292650659, 7.86183836477138691924054528738, 8.693753344415080044023494634834, 8.739180470391774304001035876449, 9.513967122264208968458849986444, 9.619147841225628573576640357938