L(s) = 1 | + (−1.95 + 0.523i)3-s + (0.959 + 0.256i)5-s + (−0.292 − 2.62i)7-s + (0.941 − 0.543i)9-s + (0.505 + 1.88i)11-s + (−2.10 − 2.10i)13-s − 2.00·15-s + (2.83 − 4.91i)17-s + (0.165 − 0.616i)19-s + (1.94 + 4.98i)21-s + (5.92 − 3.42i)23-s + (−3.47 − 2.00i)25-s + (2.73 − 2.73i)27-s + (−0.207 − 0.207i)29-s + (3.94 − 6.83i)31-s + ⋯ |
L(s) = 1 | + (−1.12 + 0.302i)3-s + (0.428 + 0.114i)5-s + (−0.110 − 0.993i)7-s + (0.313 − 0.181i)9-s + (0.152 + 0.569i)11-s + (−0.583 − 0.583i)13-s − 0.518·15-s + (0.688 − 1.19i)17-s + (0.0379 − 0.141i)19-s + (0.425 + 1.08i)21-s + (1.23 − 0.713i)23-s + (−0.695 − 0.401i)25-s + (0.526 − 0.526i)27-s + (−0.0384 − 0.0384i)29-s + (0.708 − 1.22i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.499 + 0.866i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.499 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.750276 - 0.433687i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.750276 - 0.433687i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + (0.292 + 2.62i)T \) |
good | 3 | \( 1 + (1.95 - 0.523i)T + (2.59 - 1.5i)T^{2} \) |
| 5 | \( 1 + (-0.959 - 0.256i)T + (4.33 + 2.5i)T^{2} \) |
| 11 | \( 1 + (-0.505 - 1.88i)T + (-9.52 + 5.5i)T^{2} \) |
| 13 | \( 1 + (2.10 + 2.10i)T + 13iT^{2} \) |
| 17 | \( 1 + (-2.83 + 4.91i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.165 + 0.616i)T + (-16.4 - 9.5i)T^{2} \) |
| 23 | \( 1 + (-5.92 + 3.42i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (0.207 + 0.207i)T + 29iT^{2} \) |
| 31 | \( 1 + (-3.94 + 6.83i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-9.60 - 2.57i)T + (32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + 2.40iT - 41T^{2} \) |
| 43 | \( 1 + (3.65 - 3.65i)T - 43iT^{2} \) |
| 47 | \( 1 + (-0.144 - 0.250i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (2.04 + 7.62i)T + (-45.8 + 26.5i)T^{2} \) |
| 59 | \( 1 + (3.60 + 13.4i)T + (-51.0 + 29.5i)T^{2} \) |
| 61 | \( 1 + (0.969 - 3.61i)T + (-52.8 - 30.5i)T^{2} \) |
| 67 | \( 1 + (10.0 - 2.69i)T + (58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 - 11.5iT - 71T^{2} \) |
| 73 | \( 1 + (0.310 + 0.179i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-3.84 - 6.66i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-0.424 - 0.424i)T + 83iT^{2} \) |
| 89 | \( 1 + (15.2 - 8.79i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 12.2T + 97T^{2} \) |
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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
−11.01016598974507945159763097831, −9.923646068383060056015689274439, −9.747909442972234729153775908353, −8.023525531615350638577184825021, −7.08861537168506551447449592365, −6.21024963941346251450815503740, −5.13836058364277930605572884267, −4.41383761035643363663506014982, −2.77262729633410364016877452627, −0.67066227405119755444041067425,
1.48047641409919899169552739656, 3.12376543602310466320191899225, 4.82237768965681609782626005566, 5.80897153911305589114216687615, 6.19161181451536878366642875766, 7.41598982545176417618210756860, 8.675847401453261928390404791472, 9.436065282966905942782254457619, 10.50929357843500871035884083994, 11.40731728419257254287743621274