L(s) = 1 | + (0.540 − 0.841i)2-s + (−0.909 − 0.415i)3-s + (−0.415 − 0.909i)4-s + (−0.959 − 0.281i)5-s + (−0.841 + 0.540i)6-s + (−0.989 − 0.857i)7-s + (−0.989 − 0.142i)8-s + (0.654 + 0.755i)9-s + (−0.755 + 0.654i)10-s + i·12-s + (−1.25 + 0.368i)14-s + (0.755 + 0.654i)15-s + (−0.654 + 0.755i)16-s + (0.989 − 0.142i)18-s + (0.142 + 0.989i)20-s + (0.544 + 1.19i)21-s + ⋯ |
L(s) = 1 | + (0.540 − 0.841i)2-s + (−0.909 − 0.415i)3-s + (−0.415 − 0.909i)4-s + (−0.959 − 0.281i)5-s + (−0.841 + 0.540i)6-s + (−0.989 − 0.857i)7-s + (−0.989 − 0.142i)8-s + (0.654 + 0.755i)9-s + (−0.755 + 0.654i)10-s + i·12-s + (−1.25 + 0.368i)14-s + (0.755 + 0.654i)15-s + (−0.654 + 0.755i)16-s + (0.989 − 0.142i)18-s + (0.142 + 0.989i)20-s + (0.544 + 1.19i)21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0771 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0771 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2067359811\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2067359811\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.540 + 0.841i)T \) |
| 3 | \( 1 + (0.909 + 0.415i)T \) |
| 5 | \( 1 + (0.959 + 0.281i)T \) |
| 23 | \( 1 + (0.540 - 0.841i)T \) |
good | 7 | \( 1 + (0.989 + 0.857i)T + (0.142 + 0.989i)T^{2} \) |
| 11 | \( 1 + (-0.415 + 0.909i)T^{2} \) |
| 13 | \( 1 + (0.142 - 0.989i)T^{2} \) |
| 17 | \( 1 + (0.654 + 0.755i)T^{2} \) |
| 19 | \( 1 + (-0.654 + 0.755i)T^{2} \) |
| 29 | \( 1 + (0.512 + 0.234i)T + (0.654 + 0.755i)T^{2} \) |
| 31 | \( 1 + (0.959 + 0.281i)T^{2} \) |
| 37 | \( 1 + (0.841 - 0.540i)T^{2} \) |
| 41 | \( 1 + (0.425 - 1.45i)T + (-0.841 - 0.540i)T^{2} \) |
| 43 | \( 1 + (0.822 - 0.118i)T + (0.959 - 0.281i)T^{2} \) |
| 47 | \( 1 + 1.91iT - T^{2} \) |
| 53 | \( 1 + (0.142 + 0.989i)T^{2} \) |
| 59 | \( 1 + (-0.142 + 0.989i)T^{2} \) |
| 61 | \( 1 + (1.80 + 0.258i)T + (0.959 + 0.281i)T^{2} \) |
| 67 | \( 1 + (-0.153 + 0.239i)T + (-0.415 - 0.909i)T^{2} \) |
| 71 | \( 1 + (0.415 + 0.909i)T^{2} \) |
| 73 | \( 1 + (0.654 - 0.755i)T^{2} \) |
| 79 | \( 1 + (-0.142 + 0.989i)T^{2} \) |
| 83 | \( 1 + (1.89 - 0.557i)T + (0.841 - 0.540i)T^{2} \) |
| 89 | \( 1 + (0.239 + 1.66i)T + (-0.959 + 0.281i)T^{2} \) |
| 97 | \( 1 + (0.841 + 0.540i)T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.584166675364703421475575209628, −8.351347002804522782623074747336, −7.34572923714111706021192929585, −6.64101994563086956641869193516, −5.74694575100707573746964297654, −4.78419840159856018198343611475, −3.98662703945377290224973508502, −3.19332798267110644625740580479, −1.53194963384220004626898663138, −0.16310750535178392190612851082,
2.86334532072549971072530498969, 3.75335007593710293526502945629, 4.54992542439880500807577456681, 5.51546675119952056545731365164, 6.25771995001713313721848139511, 6.85525951650704172741224966896, 7.73267475066494761033846078709, 8.740293290412461839043588263160, 9.402540260296672402585777400939, 10.40707032325258403936658349264