| L(s) = 1 | + (−0.973 − 0.230i)2-s + (1.56 − 0.739i)3-s + (0.893 + 0.448i)4-s + (0.149 − 0.201i)5-s + (−1.69 + 0.358i)6-s + (1.83 + 1.20i)7-s + (−0.766 − 0.642i)8-s + (1.90 − 2.31i)9-s + (−0.192 + 0.161i)10-s + (−1.53 + 0.179i)11-s + (1.73 + 0.0416i)12-s + (−0.160 + 0.536i)13-s + (−1.51 − 1.60i)14-s + (0.0858 − 0.426i)15-s + (0.597 + 0.802i)16-s + (−0.434 − 2.46i)17-s + ⋯ |
| L(s) = 1 | + (−0.688 − 0.163i)2-s + (0.904 − 0.427i)3-s + (0.446 + 0.224i)4-s + (0.0670 − 0.0900i)5-s + (−0.691 + 0.146i)6-s + (0.695 + 0.457i)7-s + (−0.270 − 0.227i)8-s + (0.635 − 0.772i)9-s + (−0.0608 + 0.0510i)10-s + (−0.463 + 0.0541i)11-s + (0.499 + 0.0120i)12-s + (−0.0445 + 0.148i)13-s + (−0.403 − 0.428i)14-s + (0.0221 − 0.110i)15-s + (0.149 + 0.200i)16-s + (−0.105 − 0.597i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 + 0.499i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.866 + 0.499i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.11495 - 0.298357i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.11495 - 0.298357i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.973 + 0.230i)T \) |
| 3 | \( 1 + (-1.56 + 0.739i)T \) |
| good | 5 | \( 1 + (-0.149 + 0.201i)T + (-1.43 - 4.78i)T^{2} \) |
| 7 | \( 1 + (-1.83 - 1.20i)T + (2.77 + 6.42i)T^{2} \) |
| 11 | \( 1 + (1.53 - 0.179i)T + (10.7 - 2.53i)T^{2} \) |
| 13 | \( 1 + (0.160 - 0.536i)T + (-10.8 - 7.14i)T^{2} \) |
| 17 | \( 1 + (0.434 + 2.46i)T + (-15.9 + 5.81i)T^{2} \) |
| 19 | \( 1 + (-0.461 + 2.61i)T + (-17.8 - 6.49i)T^{2} \) |
| 23 | \( 1 + (-0.530 + 0.349i)T + (9.10 - 21.1i)T^{2} \) |
| 29 | \( 1 + (4.73 - 5.01i)T + (-1.68 - 28.9i)T^{2} \) |
| 31 | \( 1 + (-0.0928 - 1.59i)T + (-30.7 + 3.59i)T^{2} \) |
| 37 | \( 1 + (9.81 + 3.57i)T + (28.3 + 23.7i)T^{2} \) |
| 41 | \( 1 + (7.44 - 1.76i)T + (36.6 - 18.4i)T^{2} \) |
| 43 | \( 1 + (3.30 - 7.65i)T + (-29.5 - 31.2i)T^{2} \) |
| 47 | \( 1 + (0.468 - 8.04i)T + (-46.6 - 5.45i)T^{2} \) |
| 53 | \( 1 + (-3.36 - 5.83i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-8.40 - 0.982i)T + (57.4 + 13.6i)T^{2} \) |
| 61 | \( 1 + (-0.457 + 0.229i)T + (36.4 - 48.9i)T^{2} \) |
| 67 | \( 1 + (5.52 + 5.85i)T + (-3.89 + 66.8i)T^{2} \) |
| 71 | \( 1 + (-12.2 + 10.2i)T + (12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 + (10.6 + 8.90i)T + (12.6 + 71.8i)T^{2} \) |
| 79 | \( 1 + (-15.0 - 3.56i)T + (70.5 + 35.4i)T^{2} \) |
| 83 | \( 1 + (7.27 + 1.72i)T + (74.1 + 37.2i)T^{2} \) |
| 89 | \( 1 + (-11.9 - 10.0i)T + (15.4 + 87.6i)T^{2} \) |
| 97 | \( 1 + (9.83 + 13.2i)T + (-27.8 + 92.9i)T^{2} \) |
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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
−12.76325902971017274827917315852, −11.76831997518847206490585808245, −10.72375237403623160788775920009, −9.405755854401616291273359642971, −8.764053803992048045040668713083, −7.74962537058962419725020606305, −6.85296990276714644681684812963, −5.07918891841439800711107696594, −3.16323106380685883668107451132, −1.77024750012067989821083313495,
2.03934956695346106728769090903, 3.76489310020126840729830417693, 5.26695245496582049391488325955, 6.98437882859923122776251947180, 8.061048998319594800040307786201, 8.641827704773554398570741530554, 10.04896326122633187178464334709, 10.48189804282574824491389037176, 11.74149003639693877213214753275, 13.17344736393031605639211900366
