| L(s) = 1 | + (0.735 − 2.01i)2-s + (1.28 + 1.16i)3-s + (−2.00 − 1.68i)4-s + (−0.0677 + 0.384i)5-s + (3.29 − 1.73i)6-s + (2.64 − 0.162i)7-s + (−1.15 + 0.666i)8-s + (0.295 + 2.98i)9-s + (0.726 + 0.419i)10-s + (−4.77 + 0.842i)11-s + (−0.617 − 4.49i)12-s + (−1.96 − 5.38i)13-s + (1.61 − 5.45i)14-s + (−0.534 + 0.414i)15-s + (−0.412 − 2.34i)16-s + (−3.57 + 6.18i)17-s + ⋯ |
| L(s) = 1 | + (0.519 − 1.42i)2-s + (0.741 + 0.671i)3-s + (−1.00 − 0.841i)4-s + (−0.0303 + 0.171i)5-s + (1.34 − 0.709i)6-s + (0.998 − 0.0613i)7-s + (−0.407 + 0.235i)8-s + (0.0983 + 0.995i)9-s + (0.229 + 0.132i)10-s + (−1.44 + 0.254i)11-s + (−0.178 − 1.29i)12-s + (−0.543 − 1.49i)13-s + (0.431 − 1.45i)14-s + (−0.137 + 0.107i)15-s + (−0.103 − 0.585i)16-s + (−0.866 + 1.50i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.347 + 0.937i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.347 + 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.50486 - 1.04744i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.50486 - 1.04744i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.28 - 1.16i)T \) |
| 7 | \( 1 + (-2.64 + 0.162i)T \) |
| good | 2 | \( 1 + (-0.735 + 2.01i)T + (-1.53 - 1.28i)T^{2} \) |
| 5 | \( 1 + (0.0677 - 0.384i)T + (-4.69 - 1.71i)T^{2} \) |
| 11 | \( 1 + (4.77 - 0.842i)T + (10.3 - 3.76i)T^{2} \) |
| 13 | \( 1 + (1.96 + 5.38i)T + (-9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (3.57 - 6.18i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.398 - 0.230i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.358 + 0.427i)T + (-3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (-1.73 + 4.76i)T + (-22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (2.14 - 2.55i)T + (-5.38 - 30.5i)T^{2} \) |
| 37 | \( 1 + (1.07 - 1.85i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (0.917 - 0.334i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (-1.16 - 6.58i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (-2.83 + 2.38i)T + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 - 3.13iT - 53T^{2} \) |
| 59 | \( 1 + (-0.715 + 4.05i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (7.89 + 9.40i)T + (-10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (-6.30 + 2.29i)T + (51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (-6.08 - 3.51i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-6.55 + 3.78i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (0.818 + 0.297i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (6.78 + 2.46i)T + (63.5 + 53.3i)T^{2} \) |
| 89 | \( 1 + (2.65 + 4.60i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (10.7 - 1.90i)T + (91.1 - 33.1i)T^{2} \) |
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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
−12.59076895149857115532071471774, −10.99901502460532424703194414715, −10.66365519121444359570582515058, −9.921576971133610626982248087669, −8.435734923667671828258771972384, −7.67670910460728247809274884263, −5.26808581449591649884416513473, −4.48819981470760105100298383819, −3.13816838662811869429350069272, −2.13048135228014141274359744276,
2.35267583813215529199681144286, 4.45432620349903008587567045947, 5.32649796875219982128178173493, 6.87061311510155474795727002097, 7.39565646339919818565532482207, 8.415267139871473532468819252050, 9.139781545821978655420328674630, 10.94462682844495594315825192834, 12.12469589832128368165146939502, 13.24447498271121140580250442352
