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 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.310418363385589811653700044158, −8.347176600830065063011597708001, −7.48872302380345665634333888719, −6.68349010497452174647849092554, −6.03422985456628983837544928472, −5.59497812122266217983233952652, −3.91780095623006477603332284466, −3.46354238308780149985539804680, −2.54463815070823346425523365640, −1.44989611560682355084572092759,
0.66536986764426518868922145386, 1.61179856406093683147734750945, 2.19504831882071540426650784558, 4.28532710472405255802317161425, 5.03586534291094215728701517591, 5.78686428832881356159532531747, 6.24311846800578080268650521203, 6.89954179704674264017170366720, 8.146410568741262068808657753141, 8.342086029742726045934728175655