L(s) = 1 | + (−0.866 − 0.5i)2-s + (0.130 − 0.991i)3-s + (0.499 + 0.866i)4-s + (−0.608 + 0.793i)6-s − 0.999i·8-s + (−0.965 − 0.258i)9-s + (−1.67 − 0.965i)11-s + (0.923 − 0.382i)12-s + (−0.5 + 0.866i)16-s − 1.21·17-s + (0.707 + 0.707i)18-s + 0.261i·19-s + (0.965 + 1.67i)22-s + (−0.991 − 0.130i)24-s + (−0.5 + 0.866i)25-s + ⋯ |
L(s) = 1 | + (−0.866 − 0.5i)2-s + (0.130 − 0.991i)3-s + (0.499 + 0.866i)4-s + (−0.608 + 0.793i)6-s − 0.999i·8-s + (−0.965 − 0.258i)9-s + (−1.67 − 0.965i)11-s + (0.923 − 0.382i)12-s + (−0.5 + 0.866i)16-s − 1.21·17-s + (0.707 + 0.707i)18-s + 0.261i·19-s + (0.965 + 1.67i)22-s + (−0.991 − 0.130i)24-s + (−0.5 + 0.866i)25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0698 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0698 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.03627808980\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.03627808980\) |
\(L(1)\) |
|
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 | \( 1 + (-0.130 + 0.991i)T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (1.67 + 0.965i)T + (0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 + 1.21T + T^{2} \) |
| 19 | \( 1 - 0.261iT - T^{2} \) |
| 23 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + (-0.793 - 1.37i)T + (-0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.258 + 0.448i)T + (-0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 + (-0.991 - 1.71i)T + (-0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 - 1.98iT - T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (0.382 - 0.662i)T + (-0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 + 1.84T + T^{2} \) |
| 97 | \( 1 + (1.37 + 0.793i)T + (0.5 + 0.866i)T^{2} \) |
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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
−8.710670099773310959515922700073, −8.251787747429075378547230993550, −7.61968539472385521607059972082, −6.99498081419462640530809283123, −6.10203326944747277448140717851, −5.38931882187139872164091866702, −4.04506979293518092129440640213, −2.90759170237342334303200629344, −2.48732685871653939661117690885, −1.34336969669496277077143150571,
0.02644674351811086044867826727, 2.16865519643851876038891156561, 2.67744211479729686954756097440, 4.15695615852862548057157456897, 4.89488356995262423757408514371, 5.50704074537300522476729585210, 6.39089740773183853470398982125, 7.30152099287126395199752979298, 7.952350333682190691185708625330, 8.609170518831056607338629337566