| L(s) = 1 | + (0.540 + 0.841i)2-s + (0.843 + 1.51i)3-s + (−0.415 + 0.909i)4-s + (−0.959 + 0.281i)5-s + (−0.816 + 1.52i)6-s + (−1.57 + 1.36i)7-s + (−0.989 + 0.142i)8-s + (−1.57 + 2.55i)9-s + (−0.755 − 0.654i)10-s + (3.18 + 2.04i)11-s + (−1.72 + 0.139i)12-s + (0.684 − 0.790i)13-s + (−2.00 − 0.588i)14-s + (−1.23 − 1.21i)15-s + (−0.654 − 0.755i)16-s + (0.274 + 0.601i)17-s + ⋯ |
| L(s) = 1 | + (0.382 + 0.594i)2-s + (0.487 + 0.873i)3-s + (−0.207 + 0.454i)4-s + (−0.429 + 0.125i)5-s + (−0.333 + 0.623i)6-s + (−0.596 + 0.516i)7-s + (−0.349 + 0.0503i)8-s + (−0.525 + 0.850i)9-s + (−0.238 − 0.207i)10-s + (0.959 + 0.616i)11-s + (−0.498 + 0.0402i)12-s + (0.189 − 0.219i)13-s + (−0.535 − 0.157i)14-s + (−0.319 − 0.313i)15-s + (−0.163 − 0.188i)16-s + (0.0665 + 0.145i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 - 0.107i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.994 - 0.107i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0853054 + 1.57924i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0853054 + 1.57924i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.540 - 0.841i)T \) |
| 3 | \( 1 + (-0.843 - 1.51i)T \) |
| 5 | \( 1 + (0.959 - 0.281i)T \) |
| 23 | \( 1 + (4.76 + 0.532i)T \) |
| good | 7 | \( 1 + (1.57 - 1.36i)T + (0.996 - 6.92i)T^{2} \) |
| 11 | \( 1 + (-3.18 - 2.04i)T + (4.56 + 10.0i)T^{2} \) |
| 13 | \( 1 + (-0.684 + 0.790i)T + (-1.85 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-0.274 - 0.601i)T + (-11.1 + 12.8i)T^{2} \) |
| 19 | \( 1 + (0.675 + 0.308i)T + (12.4 + 14.3i)T^{2} \) |
| 29 | \( 1 + (6.52 - 2.97i)T + (18.9 - 21.9i)T^{2} \) |
| 31 | \( 1 + (0.806 + 5.60i)T + (-29.7 + 8.73i)T^{2} \) |
| 37 | \( 1 + (-1.03 + 3.53i)T + (-31.1 - 20.0i)T^{2} \) |
| 41 | \( 1 + (-3.40 - 11.6i)T + (-34.4 + 22.1i)T^{2} \) |
| 43 | \( 1 + (-6.27 - 0.901i)T + (41.2 + 12.1i)T^{2} \) |
| 47 | \( 1 + 2.65iT - 47T^{2} \) |
| 53 | \( 1 + (-4.00 - 4.61i)T + (-7.54 + 52.4i)T^{2} \) |
| 59 | \( 1 + (-6.87 - 5.95i)T + (8.39 + 58.3i)T^{2} \) |
| 61 | \( 1 + (-11.8 + 1.70i)T + (58.5 - 17.1i)T^{2} \) |
| 67 | \( 1 + (1.86 + 2.90i)T + (-27.8 + 60.9i)T^{2} \) |
| 71 | \( 1 + (-5.63 - 8.76i)T + (-29.4 + 64.5i)T^{2} \) |
| 73 | \( 1 + (-4.81 + 10.5i)T + (-47.8 - 55.1i)T^{2} \) |
| 79 | \( 1 + (-4.94 - 4.28i)T + (11.2 + 78.1i)T^{2} \) |
| 83 | \( 1 + (1.44 + 0.424i)T + (69.8 + 44.8i)T^{2} \) |
| 89 | \( 1 + (1.15 - 8.02i)T + (-85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (-4.53 - 15.4i)T + (-81.6 + 52.4i)T^{2} \) |
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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
−10.87385349076721563886530308543, −9.684967384748157501755841506968, −9.253822348485667381861632275992, −8.273381581360168495620570584513, −7.46417545402081119868232361200, −6.33964376457231979887239912681, −5.49656317470831193303245228258, −4.24640126517665753903816084093, −3.70922140983790974779492218405, −2.44987702099719923869190768997,
0.69640416366333110796558893443, 2.08183021508591871705883858472, 3.51649678652135049226732696537, 3.95557278992076008913649516796, 5.64017377871392600650633384003, 6.54572629028394100490416645923, 7.33597305289436224402048621360, 8.430365889511890732491669967891, 9.152591229090093291689706725971, 10.04524781911313115797056412502
