L(s) = 1 | + (−0.309 − 0.951i)2-s + (−0.809 + 0.587i)4-s + (0.987 − 0.156i)5-s + (0.809 + 0.587i)8-s + (−0.707 − 0.707i)9-s + (−0.453 − 0.891i)10-s + (1.04 + 0.0819i)13-s + (0.309 − 0.951i)16-s + (−0.0366 + 0.152i)17-s + (−0.453 + 0.891i)18-s + (−0.707 + 0.707i)20-s + (0.951 − 0.309i)25-s + (−0.243 − 1.01i)26-s + (0.965 − 1.57i)29-s − 32-s + ⋯ |
L(s) = 1 | + (−0.309 − 0.951i)2-s + (−0.809 + 0.587i)4-s + (0.987 − 0.156i)5-s + (0.809 + 0.587i)8-s + (−0.707 − 0.707i)9-s + (−0.453 − 0.891i)10-s + (1.04 + 0.0819i)13-s + (0.309 − 0.951i)16-s + (−0.0366 + 0.152i)17-s + (−0.453 + 0.891i)18-s + (−0.707 + 0.707i)20-s + (0.951 − 0.309i)25-s + (−0.243 − 1.01i)26-s + (0.965 − 1.57i)29-s − 32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.213 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.213 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9088179517\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9088179517\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.309 + 0.951i)T \) |
| 5 | \( 1 + (-0.987 + 0.156i)T \) |
| 41 | \( 1 + (0.951 - 0.309i)T \) |
good | 3 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 7 | \( 1 + (0.156 + 0.987i)T^{2} \) |
| 11 | \( 1 + (-0.891 - 0.453i)T^{2} \) |
| 13 | \( 1 + (-1.04 - 0.0819i)T + (0.987 + 0.156i)T^{2} \) |
| 17 | \( 1 + (0.0366 - 0.152i)T + (-0.891 - 0.453i)T^{2} \) |
| 19 | \( 1 + (0.156 + 0.987i)T^{2} \) |
| 23 | \( 1 + (-0.587 - 0.809i)T^{2} \) |
| 29 | \( 1 + (-0.965 + 1.57i)T + (-0.453 - 0.891i)T^{2} \) |
| 31 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 37 | \( 1 + (0.183 - 1.16i)T + (-0.951 - 0.309i)T^{2} \) |
| 43 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 47 | \( 1 + (-0.156 + 0.987i)T^{2} \) |
| 53 | \( 1 + (1.47 - 0.355i)T + (0.891 - 0.453i)T^{2} \) |
| 59 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 61 | \( 1 + (0.278 - 0.142i)T + (0.587 - 0.809i)T^{2} \) |
| 67 | \( 1 + (-0.453 - 0.891i)T^{2} \) |
| 71 | \( 1 + (0.891 + 0.453i)T^{2} \) |
| 73 | \( 1 - 1.97iT - T^{2} \) |
| 79 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 83 | \( 1 - iT^{2} \) |
| 89 | \( 1 + (0.0600 + 0.763i)T + (-0.987 + 0.156i)T^{2} \) |
| 97 | \( 1 + (1.01 - 1.65i)T + (-0.453 - 0.891i)T^{2} \) |
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
−10.12887916109708358454323662838, −9.601121121429279537621486445146, −8.665119732575540643950911013321, −8.219142293589197658586500312704, −6.63193702142020376304542314024, −5.85870483155783788970824790394, −4.74679489281976994680127300412, −3.56379226551010250209223629259, −2.57124118521226436510873457139, −1.27181600472991252701223848036,
1.63097389078000598161265463394, 3.19283982782157259700036785662, 4.74445791954053767548113811153, 5.53916357185026516771396152034, 6.25112226449356288804314478573, 7.08860552734503423926378665538, 8.184014396836088826880635672376, 8.822549529279414544302442122969, 9.560592906656832455784713904717, 10.64234902796131964958476024534