| L(s) = 1 | + (−0.0305 − 0.173i)2-s + (−0.226 + 1.71i)3-s + (1.85 − 0.673i)4-s + (2.52 − 0.919i)5-s + (0.304 − 0.0131i)6-s + (−1.78 − 1.94i)7-s + (−0.349 − 0.605i)8-s + (−2.89 − 0.778i)9-s + (−0.236 − 0.409i)10-s + (5.66 + 2.06i)11-s + (0.736 + 3.32i)12-s + (−5.70 + 2.07i)13-s + (−0.283 + 0.369i)14-s + (1.00 + 4.54i)15-s + (2.92 − 2.45i)16-s + (−0.621 − 1.07i)17-s + ⋯ |
| L(s) = 1 | + (−0.0216 − 0.122i)2-s + (−0.130 + 0.991i)3-s + (0.925 − 0.336i)4-s + (1.12 − 0.411i)5-s + (0.124 − 0.00538i)6-s + (−0.676 − 0.736i)7-s + (−0.123 − 0.213i)8-s + (−0.965 − 0.259i)9-s + (−0.0748 − 0.129i)10-s + (1.70 + 0.621i)11-s + (0.212 + 0.961i)12-s + (−1.58 + 0.575i)13-s + (−0.0756 + 0.0988i)14-s + (0.259 + 1.17i)15-s + (0.730 − 0.613i)16-s + (−0.150 − 0.261i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 - 0.168i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.985 - 0.168i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.44336 + 0.122619i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.44336 + 0.122619i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.226 - 1.71i)T \) |
| 7 | \( 1 + (1.78 + 1.94i)T \) |
| good | 2 | \( 1 + (0.0305 + 0.173i)T + (-1.87 + 0.684i)T^{2} \) |
| 5 | \( 1 + (-2.52 + 0.919i)T + (3.83 - 3.21i)T^{2} \) |
| 11 | \( 1 + (-5.66 - 2.06i)T + (8.42 + 7.07i)T^{2} \) |
| 13 | \( 1 + (5.70 - 2.07i)T + (9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (0.621 + 1.07i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.47 - 2.54i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.622 - 3.53i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (1.80 + 0.656i)T + (22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (-0.351 + 0.128i)T + (23.7 - 19.9i)T^{2} \) |
| 37 | \( 1 + 4.85T + 37T^{2} \) |
| 41 | \( 1 + (8.12 - 2.95i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (1.51 + 8.58i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (-1.43 - 0.523i)T + (36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 + (-4.07 + 7.05i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-3.18 - 2.67i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (3.48 + 1.26i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (-0.699 + 3.96i)T + (-62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (0.161 - 0.279i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 - 4.85T + 73T^{2} \) |
| 79 | \( 1 + (0.357 + 2.02i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (-9.46 - 3.44i)T + (63.5 + 53.3i)T^{2} \) |
| 89 | \( 1 + (-2.13 + 3.69i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (0.938 + 5.32i)T + (-91.1 + 33.1i)T^{2} \) |
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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
−12.28112456532387597897603816004, −11.64093017215696043798498730515, −10.27256770578488302767578446128, −9.782451787546293394330608360222, −9.187959808735388569196527842300, −7.10480767540672641253825862450, −6.31084074609570280478816593822, −5.10060192135181017990381413073, −3.72715638597453968056260507303, −1.94444718816797377882611982119,
2.03144051275642879725203723011, 2.98730201695593688638685048169, 5.63418441487359616409847698931, 6.47614795378530491883206037936, 6.96206108370384458057467527566, 8.439490630126390390598054532999, 9.494528293613404494149144422221, 10.70319437842496648762417774849, 11.89350168656394034616830782887, 12.35597609700139053196588084127
