L(s) = 1 | + (0.5 − 0.866i)3-s + (0.5 − 0.866i)4-s − 5-s + (−0.499 − 0.866i)9-s + (0.5 + 0.866i)11-s + (−0.499 − 0.866i)12-s + (−0.5 + 0.866i)15-s + (−0.499 − 0.866i)16-s + (−0.5 + 0.866i)20-s + 25-s − 0.999·27-s + (−0.5 + 0.866i)31-s + 0.999·33-s − 0.999·36-s + 1.73i·37-s + ⋯ |
L(s) = 1 | + (0.5 − 0.866i)3-s + (0.5 − 0.866i)4-s − 5-s + (−0.499 − 0.866i)9-s + (0.5 + 0.866i)11-s + (−0.499 − 0.866i)12-s + (−0.5 + 0.866i)15-s + (−0.499 − 0.866i)16-s + (−0.5 + 0.866i)20-s + 25-s − 0.999·27-s + (−0.5 + 0.866i)31-s + 0.999·33-s − 0.999·36-s + 1.73i·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9641837252\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9641837252\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.5 + 0.866i)T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 + (-0.5 - 0.866i)T \) |
good | 2 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 7 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 - 1.73iT - T^{2} \) |
| 41 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 - 1.73iT - T^{2} \) |
| 59 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + T + T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 + 2T + T^{2} \) |
| 97 | \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \) |
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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
−11.11521181116597775839978150928, −10.08766332485277683016198908773, −9.092682472852271583318324123004, −8.176482261845682395311412400722, −7.09519838802787923575196761394, −6.76444800385854424179163979128, −5.42711059721852595299903059137, −4.10317969596452681721197654326, −2.73979297427601934793450303892, −1.37750165796056378464192362254,
2.60474746297423955927788883152, 3.67418334176520786447694638611, 4.19767395195386211538079036987, 5.70352430346461149741072132080, 7.07623465675912361294550618644, 7.906359913347076330722224531214, 8.608478768180828357330227126767, 9.370919337894258482164587114333, 10.81215844580237680779875477265, 11.18339328607365564339684216662