| L(s) = 1 | + (−1.26 + 0.635i)2-s + (1.08 + 1.16i)3-s + (1.19 − 1.60i)4-s + (−0.529 + 1.71i)5-s + (−2.11 − 0.786i)6-s + (−2.34 + 1.22i)7-s + (−0.484 + 2.78i)8-s + (0.0343 − 0.458i)9-s + (−0.422 − 2.50i)10-s + (−4.71 + 0.353i)11-s + (3.16 − 0.348i)12-s + (−0.558 + 1.15i)13-s + (2.18 − 3.03i)14-s + (−2.57 + 1.24i)15-s + (−1.15 − 3.82i)16-s + (−3.60 − 0.543i)17-s + ⋯ |
| L(s) = 1 | + (−0.893 + 0.449i)2-s + (0.625 + 0.674i)3-s + (0.595 − 0.803i)4-s + (−0.236 + 0.767i)5-s + (−0.862 − 0.321i)6-s + (−0.887 + 0.461i)7-s + (−0.171 + 0.985i)8-s + (0.0114 − 0.152i)9-s + (−0.133 − 0.792i)10-s + (−1.42 + 0.106i)11-s + (0.914 − 0.100i)12-s + (−0.154 + 0.321i)13-s + (0.585 − 0.810i)14-s + (−0.665 + 0.320i)15-s + (−0.289 − 0.957i)16-s + (−0.874 − 0.131i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0277i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0277i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.00782624 - 0.563270i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00782624 - 0.563270i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.26 - 0.635i)T \) |
| 7 | \( 1 + (2.34 - 1.22i)T \) |
| good | 3 | \( 1 + (-1.08 - 1.16i)T + (-0.224 + 2.99i)T^{2} \) |
| 5 | \( 1 + (0.529 - 1.71i)T + (-4.13 - 2.81i)T^{2} \) |
| 11 | \( 1 + (4.71 - 0.353i)T + (10.8 - 1.63i)T^{2} \) |
| 13 | \( 1 + (0.558 - 1.15i)T + (-8.10 - 10.1i)T^{2} \) |
| 17 | \( 1 + (3.60 + 0.543i)T + (16.2 + 5.01i)T^{2} \) |
| 19 | \( 1 + (-4.74 - 2.73i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (7.55 - 1.13i)T + (21.9 - 6.77i)T^{2} \) |
| 29 | \( 1 + (-6.72 - 5.36i)T + (6.45 + 28.2i)T^{2} \) |
| 31 | \( 1 + (0.368 + 0.638i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (6.57 - 2.58i)T + (27.1 - 25.1i)T^{2} \) |
| 41 | \( 1 + (1.11 + 4.87i)T + (-36.9 + 17.7i)T^{2} \) |
| 43 | \( 1 + (0.167 + 0.0381i)T + (38.7 + 18.6i)T^{2} \) |
| 47 | \( 1 + (8.06 - 5.49i)T + (17.1 - 43.7i)T^{2} \) |
| 53 | \( 1 + (-1.16 - 0.457i)T + (38.8 + 36.0i)T^{2} \) |
| 59 | \( 1 + (0.0541 + 0.175i)T + (-48.7 + 33.2i)T^{2} \) |
| 61 | \( 1 + (-6.56 + 2.57i)T + (44.7 - 41.4i)T^{2} \) |
| 67 | \( 1 + (10.3 - 5.97i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-8.89 - 11.1i)T + (-15.7 + 69.2i)T^{2} \) |
| 73 | \( 1 + (-3.14 - 2.14i)T + (26.6 + 67.9i)T^{2} \) |
| 79 | \( 1 + (0.347 - 0.601i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-5.85 - 12.1i)T + (-51.7 + 64.8i)T^{2} \) |
| 89 | \( 1 + (-0.439 + 5.86i)T + (-88.0 - 13.2i)T^{2} \) |
| 97 | \( 1 - 14.1T + 97T^{2} \) |
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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
−11.49908328458438957753559789181, −10.26251866055300242317055872760, −10.05043233120495705313379209039, −9.042743552574169697928560361224, −8.226043175700577114554568452335, −7.18120207292704020816301829674, −6.33593239459259168763030540807, −5.13301710026009398978766634877, −3.41246610794549560716808053494, −2.48547879259669051630636327296,
0.42563176802347542030321082741, 2.21451460087714046422780809690, 3.22431428670852802571350667216, 4.78386543244047260584959309559, 6.45529153535157947026736625659, 7.51896165477963359135472788740, 8.106501252323749129762256220086, 8.854241947467771858812307328548, 9.996749357106945915926453527895, 10.55417909184193849702770019553
