L(s) = 1 | + (0.410 − 0.315i)3-s + (0.965 + 0.258i)4-s + (−1.18 + 0.583i)5-s + (−0.189 + 0.707i)9-s + (0.130 + 0.991i)11-s + (0.478 − 0.198i)12-s + (−0.301 + 0.612i)15-s + (0.866 + 0.499i)16-s + (−1.29 + 0.257i)20-s + (0.665 − 0.583i)23-s + (0.449 − 0.586i)25-s + (0.343 + 0.828i)27-s + (−0.465 − 0.607i)31-s + (0.366 + 0.366i)33-s + (−0.366 + 0.633i)36-s + (1.31 − 1.50i)37-s + ⋯ |
L(s) = 1 | + (0.410 − 0.315i)3-s + (0.965 + 0.258i)4-s + (−1.18 + 0.583i)5-s + (−0.189 + 0.707i)9-s + (0.130 + 0.991i)11-s + (0.478 − 0.198i)12-s + (−0.301 + 0.612i)15-s + (0.866 + 0.499i)16-s + (−1.29 + 0.257i)20-s + (0.665 − 0.583i)23-s + (0.449 − 0.586i)25-s + (0.343 + 0.828i)27-s + (−0.465 − 0.607i)31-s + (0.366 + 0.366i)33-s + (−0.366 + 0.633i)36-s + (1.31 − 1.50i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1067 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.714 - 0.699i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1067 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.714 - 0.699i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.186356176\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.186356176\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 + (-0.130 - 0.991i)T \) |
| 97 | \( 1 + (-0.991 - 0.130i)T \) |
good | 2 | \( 1 + (-0.965 - 0.258i)T^{2} \) |
| 3 | \( 1 + (-0.410 + 0.315i)T + (0.258 - 0.965i)T^{2} \) |
| 5 | \( 1 + (1.18 - 0.583i)T + (0.608 - 0.793i)T^{2} \) |
| 7 | \( 1 + (0.991 - 0.130i)T^{2} \) |
| 13 | \( 1 + (0.608 - 0.793i)T^{2} \) |
| 17 | \( 1 + (-0.991 - 0.130i)T^{2} \) |
| 19 | \( 1 + (-0.382 - 0.923i)T^{2} \) |
| 23 | \( 1 + (-0.665 + 0.583i)T + (0.130 - 0.991i)T^{2} \) |
| 29 | \( 1 + (-0.793 - 0.608i)T^{2} \) |
| 31 | \( 1 + (0.465 + 0.607i)T + (-0.258 + 0.965i)T^{2} \) |
| 37 | \( 1 + (-1.31 + 1.50i)T + (-0.130 - 0.991i)T^{2} \) |
| 41 | \( 1 + (0.793 + 0.608i)T^{2} \) |
| 43 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 47 | \( 1 + (0.707 - 0.707i)T - iT^{2} \) |
| 53 | \( 1 + (-0.158 + 1.20i)T + (-0.965 - 0.258i)T^{2} \) |
| 59 | \( 1 + (1.13 + 0.991i)T + (0.130 + 0.991i)T^{2} \) |
| 61 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.357 + 0.534i)T + (-0.382 - 0.923i)T^{2} \) |
| 71 | \( 1 + (-0.0420 - 0.123i)T + (-0.793 + 0.608i)T^{2} \) |
| 73 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 79 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 83 | \( 1 + (0.991 + 0.130i)T^{2} \) |
| 89 | \( 1 + (0.707 + 0.707i)T^{2} \) |
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
−10.42422717828339876986867140323, −9.324129354930684522894584623172, −8.133346584732183973505856673327, −7.66522232873818150004097850544, −7.11951171448649727179978642667, −6.28025562132675441518871715236, −4.89279679117309501739128545588, −3.80064332523287782126324230810, −2.84739695623538615850853776995, −1.95563620545801859067888576266,
1.13133748402568996364604023283, 2.95347782456063275166243685701, 3.55649021007387176644791430962, 4.66025105523746843397473336212, 5.82001659626490439325347799950, 6.65151353011311174082775014160, 7.62655541195024249145186858367, 8.339260083780791468720131726850, 9.049509967897149613260872073652, 9.968870007791966219758939758351