| L(s) = 1 | + (−0.662 − 0.354i)3-s + (1.65 − 2.02i)5-s + (−2.34 − 1.56i)7-s + (−1.35 − 2.02i)9-s + (−2.03 − 0.616i)11-s + (−4.90 + 4.02i)13-s + (−1.81 + 0.752i)15-s + (1.80 + 0.746i)17-s + (−0.242 + 2.46i)19-s + (1.00 + 1.87i)21-s + (−1.97 − 0.392i)23-s + (−0.358 − 1.80i)25-s + (0.400 + 4.06i)27-s + (−2.60 − 8.58i)29-s + (6.20 − 6.20i)31-s + ⋯ |
| L(s) = 1 | + (−0.382 − 0.204i)3-s + (0.741 − 0.904i)5-s + (−0.887 − 0.593i)7-s + (−0.450 − 0.674i)9-s + (−0.613 − 0.185i)11-s + (−1.35 + 1.11i)13-s + (−0.468 + 0.194i)15-s + (0.436 + 0.180i)17-s + (−0.0556 + 0.565i)19-s + (0.218 + 0.408i)21-s + (−0.411 − 0.0817i)23-s + (−0.0717 − 0.360i)25-s + (0.0770 + 0.782i)27-s + (−0.483 − 1.59i)29-s + (1.11 − 1.11i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.918 + 0.395i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.918 + 0.395i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.131351 - 0.636939i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.131351 - 0.636939i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.662 + 0.354i)T + (1.66 + 2.49i)T^{2} \) |
| 5 | \( 1 + (-1.65 + 2.02i)T + (-0.975 - 4.90i)T^{2} \) |
| 7 | \( 1 + (2.34 + 1.56i)T + (2.67 + 6.46i)T^{2} \) |
| 11 | \( 1 + (2.03 + 0.616i)T + (9.14 + 6.11i)T^{2} \) |
| 13 | \( 1 + (4.90 - 4.02i)T + (2.53 - 12.7i)T^{2} \) |
| 17 | \( 1 + (-1.80 - 0.746i)T + (12.0 + 12.0i)T^{2} \) |
| 19 | \( 1 + (0.242 - 2.46i)T + (-18.6 - 3.70i)T^{2} \) |
| 23 | \( 1 + (1.97 + 0.392i)T + (21.2 + 8.80i)T^{2} \) |
| 29 | \( 1 + (2.60 + 8.58i)T + (-24.1 + 16.1i)T^{2} \) |
| 31 | \( 1 + (-6.20 + 6.20i)T - 31iT^{2} \) |
| 37 | \( 1 + (8.13 - 0.801i)T + (36.2 - 7.21i)T^{2} \) |
| 41 | \( 1 + (0.230 - 1.16i)T + (-37.8 - 15.6i)T^{2} \) |
| 43 | \( 1 + (2.87 - 1.53i)T + (23.8 - 35.7i)T^{2} \) |
| 47 | \( 1 + (-4.04 + 9.77i)T + (-33.2 - 33.2i)T^{2} \) |
| 53 | \( 1 + (-1.25 + 4.14i)T + (-44.0 - 29.4i)T^{2} \) |
| 59 | \( 1 + (-3.50 - 2.87i)T + (11.5 + 57.8i)T^{2} \) |
| 61 | \( 1 + (-0.456 + 0.853i)T + (-33.8 - 50.7i)T^{2} \) |
| 67 | \( 1 + (-4.42 + 8.28i)T + (-37.2 - 55.7i)T^{2} \) |
| 71 | \( 1 + (-0.382 + 0.573i)T + (-27.1 - 65.5i)T^{2} \) |
| 73 | \( 1 + (-6.49 + 4.33i)T + (27.9 - 67.4i)T^{2} \) |
| 79 | \( 1 + (6.04 + 14.6i)T + (-55.8 + 55.8i)T^{2} \) |
| 83 | \( 1 + (-1.78 - 0.175i)T + (81.4 + 16.1i)T^{2} \) |
| 89 | \( 1 + (-0.612 + 0.121i)T + (82.2 - 34.0i)T^{2} \) |
| 97 | \( 1 + (-8.11 + 8.11i)T - 97iT^{2} \) |
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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
−10.15036989108197442765422793808, −9.779553033651564457033482642249, −8.912473754183153411666350744367, −7.75687478034585853596887867229, −6.65909084110532412049201511660, −5.90334379221859073192241784500, −4.93905931513452076466070428935, −3.70112107659828082462410578421, −2.13471398995269921775493449479, −0.37024936181006370093313797246,
2.49989750499890528405271999159, 3.03102451730046535108782918587, 5.07181034851515555951720557284, 5.57107285927048430684006564720, 6.65782338199490686780295891673, 7.51908584628986669912018750180, 8.695640715958187210761221622749, 9.917552404961620538439974566196, 10.24531859297063522849299477126, 11.02666920183211937803980761338
