L(s) = 1 | + (−0.866 + 0.5i)2-s + (−0.707 − 0.707i)3-s + (0.499 − 0.866i)4-s + (0.965 − 1.67i)5-s + (0.965 + 0.258i)6-s + 0.999i·8-s + 1.00i·9-s + 1.93i·10-s + (−0.965 + 0.258i)12-s + (1.22 + 0.707i)13-s + (−1.86 + 0.5i)15-s + (−0.5 − 0.866i)16-s + (−0.500 − 0.866i)18-s − 0.517i·19-s + (−0.965 − 1.67i)20-s + ⋯ |
L(s) = 1 | + (−0.866 + 0.5i)2-s + (−0.707 − 0.707i)3-s + (0.499 − 0.866i)4-s + (0.965 − 1.67i)5-s + (0.965 + 0.258i)6-s + 0.999i·8-s + 1.00i·9-s + 1.93i·10-s + (−0.965 + 0.258i)12-s + (1.22 + 0.707i)13-s + (−1.86 + 0.5i)15-s + (−0.5 − 0.866i)16-s + (−0.500 − 0.866i)18-s − 0.517i·19-s + (−0.965 − 1.67i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.342 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.342 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7434913353\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7434913353\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 - 0.5i)T \) |
| 3 | \( 1 + (0.707 + 0.707i)T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (-0.965 + 1.67i)T + (-0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 + 0.517iT - T^{2} \) |
| 23 | \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.448 + 0.258i)T + (0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + iT - T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
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
−8.342086029742726045934728175655, −8.146410568741262068808657753141, −6.89954179704674264017170366720, −6.24311846800578080268650521203, −5.78686428832881356159532531747, −5.03586534291094215728701517591, −4.28532710472405255802317161425, −2.19504831882071540426650784558, −1.61179856406093683147734750945, −0.66536986764426518868922145386,
1.44989611560682355084572092759, 2.54463815070823346425523365640, 3.46354238308780149985539804680, 3.91780095623006477603332284466, 5.59497812122266217983233952652, 6.03422985456628983837544928472, 6.68349010497452174647849092554, 7.48872302380345665634333888719, 8.347176600830065063011597708001, 9.310418363385589811653700044158