L(s) = 1 | + (−0.636 + 1.26i)2-s + (−1.18 − 1.60i)4-s + (−1.27 + 1.27i)5-s + 0.0285i·7-s + (2.78 − 0.477i)8-s + (−0.798 − 2.42i)10-s + (−2.94 + 2.94i)11-s + (−1.34 − 1.34i)13-s + (−0.0360 − 0.0181i)14-s + (−1.17 + 3.82i)16-s − 1.40·17-s + (−5.46 − 5.46i)19-s + (3.56 + 0.534i)20-s + (−1.84 − 5.58i)22-s − 0.0963i·23-s + ⋯ |
L(s) = 1 | + (−0.450 + 0.892i)2-s + (−0.594 − 0.804i)4-s + (−0.570 + 0.570i)5-s + 0.0107i·7-s + (0.985 − 0.168i)8-s + (−0.252 − 0.766i)10-s + (−0.886 + 0.886i)11-s + (−0.373 − 0.373i)13-s + (−0.00963 − 0.00485i)14-s + (−0.292 + 0.956i)16-s − 0.339·17-s + (−1.25 − 1.25i)19-s + (0.797 + 0.119i)20-s + (−0.392 − 1.19i)22-s − 0.0200i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.636 + 0.771i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.636 + 0.771i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0526902 - 0.111806i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0526902 - 0.111806i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.636 - 1.26i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (1.27 - 1.27i)T - 5iT^{2} \) |
| 7 | \( 1 - 0.0285iT - 7T^{2} \) |
| 11 | \( 1 + (2.94 - 2.94i)T - 11iT^{2} \) |
| 13 | \( 1 + (1.34 + 1.34i)T + 13iT^{2} \) |
| 17 | \( 1 + 1.40T + 17T^{2} \) |
| 19 | \( 1 + (5.46 + 5.46i)T + 19iT^{2} \) |
| 23 | \( 1 + 0.0963iT - 23T^{2} \) |
| 29 | \( 1 + (6.62 + 6.62i)T + 29iT^{2} \) |
| 31 | \( 1 + 3.75T + 31T^{2} \) |
| 37 | \( 1 + (-4.36 + 4.36i)T - 37iT^{2} \) |
| 41 | \( 1 - 5.24iT - 41T^{2} \) |
| 43 | \( 1 + (5.87 - 5.87i)T - 43iT^{2} \) |
| 47 | \( 1 + 3.04T + 47T^{2} \) |
| 53 | \( 1 + (0.358 - 0.358i)T - 53iT^{2} \) |
| 59 | \( 1 + (4.94 - 4.94i)T - 59iT^{2} \) |
| 61 | \( 1 + (-6.84 - 6.84i)T + 61iT^{2} \) |
| 67 | \( 1 + (-1.56 - 1.56i)T + 67iT^{2} \) |
| 71 | \( 1 + 13.1iT - 71T^{2} \) |
| 73 | \( 1 - 7.44iT - 73T^{2} \) |
| 79 | \( 1 + 2.58T + 79T^{2} \) |
| 83 | \( 1 + (7.92 + 7.92i)T + 83iT^{2} \) |
| 89 | \( 1 - 9.64iT - 89T^{2} \) |
| 97 | \( 1 - 2.24T + 97T^{2} \) |
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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
−11.41822397894426953145087881704, −10.71158719809163658215363582004, −9.831982183216638399070841624362, −8.926813580996036838666187849052, −7.78329153104663106089253448570, −7.32094866028286193598965053118, −6.32357130959576449733433342715, −5.12466079985655896682015204993, −4.16181098573831416092317985908, −2.37734666260372283637850219763,
0.086418988376516679424143304024, 1.96037527342332259296561320536, 3.42732433222116303047817897913, 4.40007798940752480367733588102, 5.56039369347106107044713162256, 7.14751594560287148036994154596, 8.267353001447352883822963874597, 8.612449382054653223156488232476, 9.781127044447428359890528815346, 10.70150795879116381453364769869