| L(s) = 1 | + (0.467 − 0.249i)3-s + (0.246 + 0.300i)5-s + (−1.89 + 1.26i)7-s + (−1.51 + 2.26i)9-s + (−5.42 + 1.64i)11-s + (5.51 + 4.52i)13-s + (0.189 + 0.0786i)15-s + (−3.03 + 1.25i)17-s + (0.249 + 2.53i)19-s + (−0.567 + 1.06i)21-s + (5.91 − 1.17i)23-s + (0.946 − 4.75i)25-s + (−0.296 + 3.01i)27-s + (−0.110 + 0.363i)29-s + (0.158 + 0.158i)31-s + ⋯ |
| L(s) = 1 | + (0.269 − 0.144i)3-s + (0.110 + 0.134i)5-s + (−0.714 + 0.477i)7-s + (−0.503 + 0.753i)9-s + (−1.63 + 0.496i)11-s + (1.53 + 1.25i)13-s + (0.0490 + 0.0203i)15-s + (−0.735 + 0.304i)17-s + (0.0572 + 0.580i)19-s + (−0.123 + 0.231i)21-s + (1.23 − 0.245i)23-s + (0.189 − 0.951i)25-s + (−0.0571 + 0.580i)27-s + (−0.0204 + 0.0674i)29-s + (0.0284 + 0.0284i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0948 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0948 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.741973 + 0.816048i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.741973 + 0.816048i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 + (-0.467 + 0.249i)T + (1.66 - 2.49i)T^{2} \) |
| 5 | \( 1 + (-0.246 - 0.300i)T + (-0.975 + 4.90i)T^{2} \) |
| 7 | \( 1 + (1.89 - 1.26i)T + (2.67 - 6.46i)T^{2} \) |
| 11 | \( 1 + (5.42 - 1.64i)T + (9.14 - 6.11i)T^{2} \) |
| 13 | \( 1 + (-5.51 - 4.52i)T + (2.53 + 12.7i)T^{2} \) |
| 17 | \( 1 + (3.03 - 1.25i)T + (12.0 - 12.0i)T^{2} \) |
| 19 | \( 1 + (-0.249 - 2.53i)T + (-18.6 + 3.70i)T^{2} \) |
| 23 | \( 1 + (-5.91 + 1.17i)T + (21.2 - 8.80i)T^{2} \) |
| 29 | \( 1 + (0.110 - 0.363i)T + (-24.1 - 16.1i)T^{2} \) |
| 31 | \( 1 + (-0.158 - 0.158i)T + 31iT^{2} \) |
| 37 | \( 1 + (4.28 + 0.421i)T + (36.2 + 7.21i)T^{2} \) |
| 41 | \( 1 + (1.22 + 6.16i)T + (-37.8 + 15.6i)T^{2} \) |
| 43 | \( 1 + (-7.59 - 4.05i)T + (23.8 + 35.7i)T^{2} \) |
| 47 | \( 1 + (1.45 + 3.51i)T + (-33.2 + 33.2i)T^{2} \) |
| 53 | \( 1 + (-2.06 - 6.80i)T + (-44.0 + 29.4i)T^{2} \) |
| 59 | \( 1 + (7.12 - 5.84i)T + (11.5 - 57.8i)T^{2} \) |
| 61 | \( 1 + (-2.08 - 3.90i)T + (-33.8 + 50.7i)T^{2} \) |
| 67 | \( 1 + (1.02 + 1.92i)T + (-37.2 + 55.7i)T^{2} \) |
| 71 | \( 1 + (-2.79 - 4.17i)T + (-27.1 + 65.5i)T^{2} \) |
| 73 | \( 1 + (8.95 + 5.98i)T + (27.9 + 67.4i)T^{2} \) |
| 79 | \( 1 + (-3.96 + 9.56i)T + (-55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 + (-10.9 + 1.07i)T + (81.4 - 16.1i)T^{2} \) |
| 89 | \( 1 + (2.99 + 0.594i)T + (82.2 + 34.0i)T^{2} \) |
| 97 | \( 1 + (-4.90 - 4.90i)T + 97iT^{2} \) |
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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
−10.83740187555395746160459944068, −10.51748053812972208422428918439, −9.123670868213220103585169432281, −8.598359696296723452084203133106, −7.57280618098692993626580215573, −6.50855165450734187668474729510, −5.63754942777923079715044868789, −4.46952638350931992072764949511, −3.03136129086701910695670690899, −2.06530673329172135683690598873,
0.61436682167222470076323999726, 2.92780917564644405654875146208, 3.48239215162964482041915537879, 5.10762173970303956401590227404, 5.96175269518258744508523171940, 6.99516143804902094711991240508, 8.139977022939798816711218467953, 8.840492188708271128376147648921, 9.740457610675171298216920995662, 10.83422856837897996831659831160
