| L(s) = 1 | + (−1.19 + 0.762i)2-s + (−0.209 − 1.39i)3-s + (0.836 − 1.81i)4-s + (1.73 + 0.679i)5-s + (1.31 + 1.49i)6-s + (−0.592 − 2.57i)7-s + (0.389 + 2.80i)8-s + (0.971 − 0.299i)9-s + (−2.58 + 0.511i)10-s + (−0.224 + 0.728i)11-s + (−2.70 − 0.783i)12-s + (1.74 − 0.398i)13-s + (2.67 + 2.61i)14-s + (0.582 − 2.55i)15-s + (−2.60 − 3.03i)16-s + (4.21 − 2.87i)17-s + ⋯ |
| L(s) = 1 | + (−0.842 + 0.539i)2-s + (−0.121 − 0.803i)3-s + (0.418 − 0.908i)4-s + (0.774 + 0.303i)5-s + (0.535 + 0.611i)6-s + (−0.223 − 0.974i)7-s + (0.137 + 0.990i)8-s + (0.323 − 0.0999i)9-s + (−0.815 + 0.161i)10-s + (−0.0677 + 0.219i)11-s + (−0.780 − 0.226i)12-s + (0.483 − 0.110i)13-s + (0.714 + 0.700i)14-s + (0.150 − 0.659i)15-s + (−0.650 − 0.759i)16-s + (1.02 − 0.697i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.591 + 0.806i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.591 + 0.806i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.894829 - 0.453668i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.894829 - 0.453668i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.19 - 0.762i)T \) |
| 7 | \( 1 + (0.592 + 2.57i)T \) |
| good | 3 | \( 1 + (0.209 + 1.39i)T + (-2.86 + 0.884i)T^{2} \) |
| 5 | \( 1 + (-1.73 - 0.679i)T + (3.66 + 3.40i)T^{2} \) |
| 11 | \( 1 + (0.224 - 0.728i)T + (-9.08 - 6.19i)T^{2} \) |
| 13 | \( 1 + (-1.74 + 0.398i)T + (11.7 - 5.64i)T^{2} \) |
| 17 | \( 1 + (-4.21 + 2.87i)T + (6.21 - 15.8i)T^{2} \) |
| 19 | \( 1 + (7.23 - 4.17i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.115 + 0.0785i)T + (8.40 + 21.4i)T^{2} \) |
| 29 | \( 1 + (-2.76 + 5.74i)T + (-18.0 - 22.6i)T^{2} \) |
| 31 | \( 1 + (-3.93 + 6.81i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-4.03 + 0.302i)T + (36.5 - 5.51i)T^{2} \) |
| 41 | \( 1 + (-2.31 - 2.90i)T + (-9.12 + 39.9i)T^{2} \) |
| 43 | \( 1 + (-6.71 - 5.35i)T + (9.56 + 41.9i)T^{2} \) |
| 47 | \( 1 + (-4.15 + 3.85i)T + (3.51 - 46.8i)T^{2} \) |
| 53 | \( 1 + (7.02 + 0.526i)T + (52.4 + 7.89i)T^{2} \) |
| 59 | \( 1 + (7.74 - 3.03i)T + (43.2 - 40.1i)T^{2} \) |
| 61 | \( 1 + (7.47 - 0.559i)T + (60.3 - 9.09i)T^{2} \) |
| 67 | \( 1 + (-6.85 - 3.95i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-2.31 + 1.11i)T + (44.2 - 55.5i)T^{2} \) |
| 73 | \( 1 + (-0.0710 - 0.0659i)T + (5.45 + 72.7i)T^{2} \) |
| 79 | \( 1 + (2.79 + 4.84i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-9.51 - 2.17i)T + (74.7 + 36.0i)T^{2} \) |
| 89 | \( 1 + (0.741 - 0.228i)T + (73.5 - 50.1i)T^{2} \) |
| 97 | \( 1 + 12.3T + 97T^{2} \) |
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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
−10.84513998904672491337664916354, −10.06271743632664919205128385774, −9.563084494615403755335753237781, −8.019139070678468607852075116639, −7.56809964265498397334140332094, −6.29726165461507487688741154755, −6.15157816190200477069250577807, −4.31651849007946742435227110507, −2.28517749118359507128185713897, −0.963507012362228674191529799460,
1.69578402138929601517009304343, 3.06042347492544007118475289949, 4.40065472081456922864650716554, 5.67847076357748437579379331146, 6.70269731461709644850447726498, 8.187133118466248652551518711189, 9.004663313535094650645186675292, 9.550560454621005295111159740821, 10.55105168670314744059697724709, 10.97635563492406756416653573332
