| L(s) = 1 | + (−0.142 + 0.989i)2-s + (0.857 + 0.989i)3-s + (−0.959 − 0.281i)4-s + (−0.841 − 0.540i)5-s + (−1.10 + 0.708i)6-s + (0.408 − 2.84i)7-s + (0.415 − 0.909i)8-s + (0.182 − 1.27i)9-s + (0.654 − 0.755i)10-s + (−2.20 − 1.41i)11-s + (−0.544 − 1.19i)12-s + (0.366 + 0.803i)13-s + (2.75 + 0.808i)14-s + (−0.186 − 1.29i)15-s + (0.841 + 0.540i)16-s + (2.90 − 0.853i)17-s + ⋯ |
| L(s) = 1 | + (−0.100 + 0.699i)2-s + (0.495 + 0.571i)3-s + (−0.479 − 0.140i)4-s + (−0.376 − 0.241i)5-s + (−0.449 + 0.289i)6-s + (0.154 − 1.07i)7-s + (0.146 − 0.321i)8-s + (0.0609 − 0.423i)9-s + (0.207 − 0.238i)10-s + (−0.664 − 0.426i)11-s + (−0.157 − 0.343i)12-s + (0.101 + 0.222i)13-s + (0.735 + 0.216i)14-s + (−0.0481 − 0.334i)15-s + (0.210 + 0.135i)16-s + (0.704 − 0.206i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.930 + 0.365i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.930 + 0.365i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.27519 - 0.241733i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.27519 - 0.241733i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.142 - 0.989i)T \) |
| 5 | \( 1 + (0.841 + 0.540i)T \) |
| 67 | \( 1 + (1.41 + 8.06i)T \) |
| good | 3 | \( 1 + (-0.857 - 0.989i)T + (-0.426 + 2.96i)T^{2} \) |
| 7 | \( 1 + (-0.408 + 2.84i)T + (-6.71 - 1.97i)T^{2} \) |
| 11 | \( 1 + (2.20 + 1.41i)T + (4.56 + 10.0i)T^{2} \) |
| 13 | \( 1 + (-0.366 - 0.803i)T + (-8.51 + 9.82i)T^{2} \) |
| 17 | \( 1 + (-2.90 + 0.853i)T + (14.3 - 9.19i)T^{2} \) |
| 19 | \( 1 + (0.724 + 5.03i)T + (-18.2 + 5.35i)T^{2} \) |
| 23 | \( 1 + (3.91 + 4.52i)T + (-3.27 + 22.7i)T^{2} \) |
| 29 | \( 1 - 2.40T + 29T^{2} \) |
| 31 | \( 1 + (0.0561 - 0.122i)T + (-20.3 - 23.4i)T^{2} \) |
| 37 | \( 1 - 1.74T + 37T^{2} \) |
| 41 | \( 1 + (5.21 - 1.52i)T + (34.4 - 22.1i)T^{2} \) |
| 43 | \( 1 + (-5.41 + 1.59i)T + (36.1 - 23.2i)T^{2} \) |
| 47 | \( 1 + (-8.55 - 9.87i)T + (-6.68 + 46.5i)T^{2} \) |
| 53 | \( 1 + (11.0 + 3.24i)T + (44.5 + 28.6i)T^{2} \) |
| 59 | \( 1 + (3.37 - 7.39i)T + (-38.6 - 44.5i)T^{2} \) |
| 61 | \( 1 + (-0.832 + 0.535i)T + (25.3 - 55.4i)T^{2} \) |
| 71 | \( 1 + (-11.6 - 3.42i)T + (59.7 + 38.3i)T^{2} \) |
| 73 | \( 1 + (3.93 - 2.52i)T + (30.3 - 66.4i)T^{2} \) |
| 79 | \( 1 + (-1.75 - 3.83i)T + (-51.7 + 59.7i)T^{2} \) |
| 83 | \( 1 + (14.2 + 9.15i)T + (34.4 + 75.4i)T^{2} \) |
| 89 | \( 1 + (0.284 - 0.328i)T + (-12.6 - 88.0i)T^{2} \) |
| 97 | \( 1 - 3.54T + 97T^{2} \) |
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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
−10.31232964110964909023333542830, −9.480998574565217111617189623753, −8.636411751418922700319958795220, −7.894181106318050410631855132337, −7.06736274207522227453975525480, −6.06870886015287614755414652299, −4.73622429332521079913932057930, −4.12999445252976670579716799593, −3.00261028602459379178012854436, −0.70054738648122189475455209337,
1.72749990266943188385837252863, 2.59954127094980639773839710668, 3.68269308674155333217324335267, 5.06140059961917907186473037245, 5.95052543760619407296684766772, 7.45505762775819508532736574558, 8.019467419897873204240428151206, 8.693828858261894421096214109621, 9.851156844913382691220446073330, 10.49897345332296080695797666805
