| L(s) = 1 | + (−1.01 + 0.981i)2-s + (−1.62 − 0.604i)3-s + (0.0714 − 1.99i)4-s + (−0.345 + 2.20i)5-s + (2.24 − 0.978i)6-s + 0.309i·7-s + (1.89 + 2.10i)8-s + (2.26 + 1.96i)9-s + (−1.81 − 2.58i)10-s + (1.53 − 1.11i)11-s + (−1.32 + 3.20i)12-s + (−3.79 − 2.75i)13-s + (−0.304 − 0.315i)14-s + (1.89 − 3.37i)15-s + (−3.98 − 0.285i)16-s + (−4.30 + 1.39i)17-s + ⋯ |
| L(s) = 1 | + (−0.719 + 0.694i)2-s + (−0.937 − 0.349i)3-s + (0.0357 − 0.999i)4-s + (−0.154 + 0.987i)5-s + (0.916 − 0.399i)6-s + 0.117i·7-s + (0.668 + 0.743i)8-s + (0.756 + 0.654i)9-s + (−0.574 − 0.818i)10-s + (0.463 − 0.336i)11-s + (−0.382 + 0.923i)12-s + (−1.05 − 0.764i)13-s + (−0.0813 − 0.0842i)14-s + (0.489 − 0.871i)15-s + (−0.997 − 0.0713i)16-s + (−1.04 + 0.339i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.934 + 0.357i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.934 + 0.357i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0234989 - 0.127258i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0234989 - 0.127258i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.01 - 0.981i)T \) |
| 3 | \( 1 + (1.62 + 0.604i)T \) |
| 5 | \( 1 + (0.345 - 2.20i)T \) |
| good | 7 | \( 1 - 0.309iT - 7T^{2} \) |
| 11 | \( 1 + (-1.53 + 1.11i)T + (3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (3.79 + 2.75i)T + (4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (4.30 - 1.39i)T + (13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (5.24 - 1.70i)T + (15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (5.20 - 3.78i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (6.49 + 2.11i)T + (23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-5.73 + 1.86i)T + (25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (-3.24 - 2.35i)T + (11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (0.902 - 1.24i)T + (-12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + 3.75iT - 43T^{2} \) |
| 47 | \( 1 + (2.19 - 6.75i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (-0.721 - 0.234i)T + (42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (11.0 + 8.03i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (7.66 - 5.57i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (-6.37 + 2.07i)T + (54.2 - 39.3i)T^{2} \) |
| 71 | \( 1 + (0.750 - 2.30i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (7.60 - 5.52i)T + (22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (1.81 + 0.588i)T + (63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (-0.792 - 2.43i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (-5.28 - 7.27i)T + (-27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (-3.88 + 11.9i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−11.96524907218830231617575660648, −11.12669418537051887098053880863, −10.41763258876685143570284166783, −9.613804472361625687449238832899, −8.133473436861875745667668585707, −7.39623298308295053655339280551, −6.38425204842107346535624809833, −5.83414723017458136680470246574, −4.36296892829688081114984881531, −2.13088821182782588319587509126,
0.12763572657619200362039436968, 1.96894881009564204983648713591, 4.20516522350938865750766111552, 4.65722721449629526475932079245, 6.39887225896680818634586947348, 7.39947266599988520292070985155, 8.758435705398103454857227573736, 9.400774992558435806412430300076, 10.29464743709863856759639410581, 11.26415201200502380136450991477
