L(s) = 1 | + (−0.309 − 0.951i)3-s + (−0.951 − 0.309i)4-s − i·5-s + (−0.809 + 0.587i)9-s + (−0.587 − 0.809i)11-s + 0.999i·12-s + (−0.951 + 0.309i)15-s + (0.809 + 0.587i)16-s + (−0.309 + 0.951i)20-s + (−1.76 − 0.278i)23-s − 25-s + (0.809 + 0.587i)27-s + (0.363 + 1.11i)31-s + (−0.587 + 0.809i)33-s + (0.951 − 0.309i)36-s + (−0.221 − 1.39i)37-s + ⋯ |
L(s) = 1 | + (−0.309 − 0.951i)3-s + (−0.951 − 0.309i)4-s − i·5-s + (−0.809 + 0.587i)9-s + (−0.587 − 0.809i)11-s + 0.999i·12-s + (−0.951 + 0.309i)15-s + (0.809 + 0.587i)16-s + (−0.309 + 0.951i)20-s + (−1.76 − 0.278i)23-s − 25-s + (0.809 + 0.587i)27-s + (0.363 + 1.11i)31-s + (−0.587 + 0.809i)33-s + (0.951 − 0.309i)36-s + (−0.221 − 1.39i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 + 0.125i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 + 0.125i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4795175474\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4795175474\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.309 + 0.951i)T \) |
| 5 | \( 1 + iT \) |
| 11 | \( 1 + (0.587 + 0.809i)T \) |
good | 2 | \( 1 + (0.951 + 0.309i)T^{2} \) |
| 7 | \( 1 - iT^{2} \) |
| 13 | \( 1 + (-0.951 + 0.309i)T^{2} \) |
| 17 | \( 1 + (-0.587 - 0.809i)T^{2} \) |
| 19 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 23 | \( 1 + (1.76 + 0.278i)T + (0.951 + 0.309i)T^{2} \) |
| 29 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 31 | \( 1 + (-0.363 - 1.11i)T + (-0.809 + 0.587i)T^{2} \) |
| 37 | \( 1 + (0.221 + 1.39i)T + (-0.951 + 0.309i)T^{2} \) |
| 41 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 43 | \( 1 + iT^{2} \) |
| 47 | \( 1 + (0.642 + 1.26i)T + (-0.587 + 0.809i)T^{2} \) |
| 53 | \( 1 + (-1.76 + 0.896i)T + (0.587 - 0.809i)T^{2} \) |
| 59 | \( 1 + (1.30 + 0.951i)T + (0.309 + 0.951i)T^{2} \) |
| 61 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 67 | \( 1 + (-0.809 + 1.58i)T + (-0.587 - 0.809i)T^{2} \) |
| 71 | \( 1 + (-1.11 - 0.363i)T + (0.809 + 0.587i)T^{2} \) |
| 73 | \( 1 + (0.951 + 0.309i)T^{2} \) |
| 79 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 83 | \( 1 + (0.587 + 0.809i)T^{2} \) |
| 89 | \( 1 + (-1.53 + 1.11i)T + (0.309 - 0.951i)T^{2} \) |
| 97 | \( 1 + (-0.278 + 0.142i)T + (0.587 - 0.809i)T^{2} \) |
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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
−9.999988787068063262373544815544, −8.940677903769696787391798573652, −8.339165320279189554666847044986, −7.73943879178065716129610778529, −6.32839715444749050715654116857, −5.55822361481768071714962103257, −4.89782099441090008358061899639, −3.66551165908730171829921417021, −1.93953211410501952469502770082, −0.51276616136996274215340704142,
2.58710438200648431660924031663, 3.72749680691850918128272554673, 4.42980121086140964642187040003, 5.44547427814806156324732161567, 6.34207034023721292147924103846, 7.59709474403710844062221017793, 8.298243689854554743561949277097, 9.470342331301778581229371093891, 9.990980069323676639454475340936, 10.52920802422778068465786180664