| L(s) = 1 | + (−1.38 − 0.282i)2-s + (−0.560 + 1.63i)3-s + (1.84 + 0.782i)4-s + (−0.963 − 1.49i)5-s + (1.23 − 2.11i)6-s + (−1.51 − 0.218i)7-s + (−2.32 − 1.60i)8-s + (−2.37 − 1.83i)9-s + (0.912 + 2.35i)10-s + (0.389 + 0.853i)11-s + (−2.31 + 2.57i)12-s + (−0.592 − 4.12i)13-s + (2.04 + 0.731i)14-s + (2.99 − 0.739i)15-s + (2.77 + 2.88i)16-s + (−1.29 − 1.12i)17-s + ⋯ |
| L(s) = 1 | + (−0.979 − 0.199i)2-s + (−0.323 + 0.946i)3-s + (0.920 + 0.391i)4-s + (−0.430 − 0.670i)5-s + (0.505 − 0.862i)6-s + (−0.573 − 0.0825i)7-s + (−0.823 − 0.567i)8-s + (−0.790 − 0.612i)9-s + (0.288 + 0.743i)10-s + (0.117 + 0.257i)11-s + (−0.667 + 0.744i)12-s + (−0.164 − 1.14i)13-s + (0.545 + 0.195i)14-s + (0.774 − 0.190i)15-s + (0.693 + 0.720i)16-s + (−0.314 − 0.272i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.206 + 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.206 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.225938 - 0.278524i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.225938 - 0.278524i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.38 + 0.282i)T \) |
| 3 | \( 1 + (0.560 - 1.63i)T \) |
| 23 | \( 1 + (1.67 + 4.49i)T \) |
| good | 5 | \( 1 + (0.963 + 1.49i)T + (-2.07 + 4.54i)T^{2} \) |
| 7 | \( 1 + (1.51 + 0.218i)T + (6.71 + 1.97i)T^{2} \) |
| 11 | \( 1 + (-0.389 - 0.853i)T + (-7.20 + 8.31i)T^{2} \) |
| 13 | \( 1 + (0.592 + 4.12i)T + (-12.4 + 3.66i)T^{2} \) |
| 17 | \( 1 + (1.29 + 1.12i)T + (2.41 + 16.8i)T^{2} \) |
| 19 | \( 1 + (-1.48 + 1.28i)T + (2.70 - 18.8i)T^{2} \) |
| 29 | \( 1 + (2.40 + 2.08i)T + (4.12 + 28.7i)T^{2} \) |
| 31 | \( 1 + (0.492 + 1.67i)T + (-26.0 + 16.7i)T^{2} \) |
| 37 | \( 1 + (4.75 + 3.05i)T + (15.3 + 33.6i)T^{2} \) |
| 41 | \( 1 + (-1.19 - 1.86i)T + (-17.0 + 37.2i)T^{2} \) |
| 43 | \( 1 + (0.604 - 2.05i)T + (-36.1 - 23.2i)T^{2} \) |
| 47 | \( 1 + 11.1T + 47T^{2} \) |
| 53 | \( 1 + (-13.0 - 1.87i)T + (50.8 + 14.9i)T^{2} \) |
| 59 | \( 1 + (-1.78 - 12.4i)T + (-56.6 + 16.6i)T^{2} \) |
| 61 | \( 1 + (-8.79 + 2.58i)T + (51.3 - 32.9i)T^{2} \) |
| 67 | \( 1 + (9.23 + 4.21i)T + (43.8 + 50.6i)T^{2} \) |
| 71 | \( 1 + (-1.24 + 2.71i)T + (-46.4 - 53.6i)T^{2} \) |
| 73 | \( 1 + (-4.83 - 5.58i)T + (-10.3 + 72.2i)T^{2} \) |
| 79 | \( 1 + (13.5 - 1.94i)T + (75.7 - 22.2i)T^{2} \) |
| 83 | \( 1 + (5.56 + 3.57i)T + (34.4 + 75.4i)T^{2} \) |
| 89 | \( 1 + (2.03 - 6.93i)T + (-74.8 - 48.1i)T^{2} \) |
| 97 | \( 1 + (5.32 - 3.42i)T + (40.2 - 88.2i)T^{2} \) |
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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
−11.45245859466651929853699445825, −10.43285985415572323906237201020, −9.809148380434760913938134080269, −8.884216390677821139886106904041, −8.076488885606563607357521378759, −6.78066926670266713183156748678, −5.54738109959791759293403030096, −4.18467155501521106147212696202, −2.89638292150090686351458193771, −0.37452583105192176819918175267,
1.78326063477413751373239258028, 3.29607960847001806853523663110, 5.55768056878872513150765796853, 6.69769738454108694419362181769, 7.10556318491752098387914578179, 8.196056967537265086591196149674, 9.187056083720586231515961758103, 10.27413487526930966363912171800, 11.39613242631148693271376506124, 11.71128461692299744762021781748
