| L(s) = 1 | + (0.158 − 0.275i)5-s + (−2.93 + 1.69i)7-s + (0.642 − 0.370i)11-s + (−2.59 − 1.5i)13-s + 4.24i·17-s + 2.57·19-s + (2.02 − 3.50i)23-s + (2.44 + 4.24i)25-s + (4.01 + 6.94i)29-s + (−4.24 − 2.45i)31-s + 1.07i·35-s − 7.34i·37-s + (0.825 + 0.476i)41-s + (−3.50 − 6.06i)43-s + (−5.15 − 8.93i)47-s + ⋯ |
| L(s) = 1 | + (0.0710 − 0.123i)5-s + (−1.10 + 0.639i)7-s + (0.193 − 0.111i)11-s + (−0.720 − 0.416i)13-s + 1.02i·17-s + 0.589·19-s + (0.421 − 0.730i)23-s + (0.489 + 0.848i)25-s + (0.745 + 1.29i)29-s + (−0.762 − 0.440i)31-s + 0.181i·35-s − 1.20i·37-s + (0.128 + 0.0744i)41-s + (−0.533 − 0.924i)43-s + (−0.752 − 1.30i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.885 + 0.463i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.885 + 0.463i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1799226575\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1799226575\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (-0.158 + 0.275i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (2.93 - 1.69i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.642 + 0.370i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (2.59 + 1.5i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 - 4.24iT - 17T^{2} \) |
| 19 | \( 1 - 2.57T + 19T^{2} \) |
| 23 | \( 1 + (-2.02 + 3.50i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-4.01 - 6.94i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (4.24 + 2.45i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 7.34iT - 37T^{2} \) |
| 41 | \( 1 + (-0.825 - 0.476i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (3.50 + 6.06i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (5.15 + 8.93i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 5.51T + 53T^{2} \) |
| 59 | \( 1 + (3.50 + 2.02i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (7.79 - 4.5i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (6.36 - 11.0i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 13.5T + 71T^{2} \) |
| 73 | \( 1 + 6.44T + 73T^{2} \) |
| 79 | \( 1 + (8.29 - 4.78i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (2.21 - 1.27i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 14.6iT - 89T^{2} \) |
| 97 | \( 1 + (5.62 + 9.74i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−8.611755846651761682294784640692, −7.29645141325187307058815539397, −6.93274738975254753697037762186, −5.85410398176732363085801538328, −5.47659422631321957715510662102, −4.41064709331463625218173949352, −3.35977630508768964368068217290, −2.79059753722903158900285715944, −1.57086379368752640300409908761, −0.05558751881907821841637125275,
1.25045524145073852638542602688, 2.71590156486951611121979863999, 3.23533564675409758391144351280, 4.36458530263601617730359525938, 4.97080075031761910199896201432, 6.11656462104768656455410985602, 6.70180374213723463625159977563, 7.32928810594224192202482525363, 8.012324804635618407023502144713, 9.195413439294269030811515788361
