L(s) = 1 | + (0.453 − 0.891i)2-s + (−0.587 − 0.809i)4-s + (−0.891 − 0.453i)5-s + (1.39 − 1.39i)7-s + (−0.987 + 0.156i)8-s + (−0.809 + 0.587i)10-s + (0.863 − 0.280i)11-s + (−0.610 − 1.87i)14-s + (−0.309 + 0.951i)16-s + (0.156 + 0.987i)20-s + (0.142 − 0.896i)22-s + (0.587 + 0.809i)25-s + (−1.95 − 0.309i)28-s + (−1.14 + 0.831i)29-s + (−0.951 − 0.690i)31-s + (0.707 + 0.707i)32-s + ⋯ |
L(s) = 1 | + (0.453 − 0.891i)2-s + (−0.587 − 0.809i)4-s + (−0.891 − 0.453i)5-s + (1.39 − 1.39i)7-s + (−0.987 + 0.156i)8-s + (−0.809 + 0.587i)10-s + (0.863 − 0.280i)11-s + (−0.610 − 1.87i)14-s + (−0.309 + 0.951i)16-s + (0.156 + 0.987i)20-s + (0.142 − 0.896i)22-s + (0.587 + 0.809i)25-s + (−1.95 − 0.309i)28-s + (−1.14 + 0.831i)29-s + (−0.951 − 0.690i)31-s + (0.707 + 0.707i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.860 + 0.509i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.860 + 0.509i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.288321654\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.288321654\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.453 + 0.891i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (0.891 + 0.453i)T \) |
good | 7 | \( 1 + (-1.39 + 1.39i)T - iT^{2} \) |
| 11 | \( 1 + (-0.863 + 0.280i)T + (0.809 - 0.587i)T^{2} \) |
| 13 | \( 1 + (-0.587 + 0.809i)T^{2} \) |
| 17 | \( 1 + (0.951 - 0.309i)T^{2} \) |
| 19 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 23 | \( 1 + (-0.587 - 0.809i)T^{2} \) |
| 29 | \( 1 + (1.14 - 0.831i)T + (0.309 - 0.951i)T^{2} \) |
| 31 | \( 1 + (0.951 + 0.690i)T + (0.309 + 0.951i)T^{2} \) |
| 37 | \( 1 + (0.587 - 0.809i)T^{2} \) |
| 41 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 43 | \( 1 - iT^{2} \) |
| 47 | \( 1 + (-0.951 - 0.309i)T^{2} \) |
| 53 | \( 1 + (-0.297 + 1.87i)T + (-0.951 - 0.309i)T^{2} \) |
| 59 | \( 1 + (0.550 - 1.69i)T + (-0.809 - 0.587i)T^{2} \) |
| 61 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 67 | \( 1 + (-0.951 + 0.309i)T^{2} \) |
| 71 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 73 | \( 1 + (0.642 - 1.26i)T + (-0.587 - 0.809i)T^{2} \) |
| 79 | \( 1 + (-1.11 - 1.53i)T + (-0.309 + 0.951i)T^{2} \) |
| 83 | \( 1 + (-1.59 + 0.253i)T + (0.951 - 0.309i)T^{2} \) |
| 89 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 97 | \( 1 + (0.0489 - 0.309i)T + (-0.951 - 0.309i)T^{2} \) |
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
−9.139571232646138537023690261777, −8.458947768421540257855065044176, −7.62356503577734419859228900525, −6.89048399857473243698078830954, −5.52382713231716710333909135926, −4.76282133606476899253619575552, −3.99204477266767637439903884486, −3.58054275344553991565452221319, −1.82901514207617851340193793968, −0.920013236065428239616935050008,
2.00161302584921422221438921037, 3.23704445615472740035383897178, 4.21888393545930830066480682473, 4.93280836760983396324035117773, 5.77776435287569298791109558144, 6.57479218654022018702834855886, 7.62021481145295059361336799458, 7.929432490396753994560279727263, 8.928078395260447889245947628349, 9.246883889261492964777244550693