| L(s) = 1 | + (−1.08 + 0.911i)2-s + (2.58 + 1.87i)3-s + (0.336 − 1.97i)4-s + (0.602 + 2.15i)5-s + (−4.50 + 0.326i)6-s − 2.97i·7-s + (1.43 + 2.43i)8-s + (2.21 + 6.82i)9-s + (−2.61 − 1.77i)10-s + (0.263 + 0.0855i)11-s + (4.56 − 4.45i)12-s + (−1.36 − 4.19i)13-s + (2.71 + 3.21i)14-s + (−2.48 + 6.68i)15-s + (−3.77 − 1.32i)16-s + (−3.01 − 4.14i)17-s + ⋯ |
| L(s) = 1 | + (−0.764 + 0.644i)2-s + (1.49 + 1.08i)3-s + (0.168 − 0.985i)4-s + (0.269 + 0.963i)5-s + (−1.83 + 0.133i)6-s − 1.12i·7-s + (0.506 + 0.861i)8-s + (0.739 + 2.27i)9-s + (−0.826 − 0.562i)10-s + (0.0794 + 0.0258i)11-s + (1.31 − 1.28i)12-s + (−0.378 − 1.16i)13-s + (0.725 + 0.860i)14-s + (−0.641 + 1.72i)15-s + (−0.943 − 0.331i)16-s + (−0.730 − 1.00i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0717 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0717 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.902232 + 0.969426i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.902232 + 0.969426i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.08 - 0.911i)T \) |
| 5 | \( 1 + (-0.602 - 2.15i)T \) |
| good | 3 | \( 1 + (-2.58 - 1.87i)T + (0.927 + 2.85i)T^{2} \) |
| 7 | \( 1 + 2.97iT - 7T^{2} \) |
| 11 | \( 1 + (-0.263 - 0.0855i)T + (8.89 + 6.46i)T^{2} \) |
| 13 | \( 1 + (1.36 + 4.19i)T + (-10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (3.01 + 4.14i)T + (-5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (-0.824 - 1.13i)T + (-5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (0.307 + 0.0998i)T + (18.6 + 13.5i)T^{2} \) |
| 29 | \( 1 + (2.59 - 3.56i)T + (-8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (3.04 - 2.21i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (1.24 + 3.84i)T + (-29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (1.97 + 6.09i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 - 10.9T + 43T^{2} \) |
| 47 | \( 1 + (-1.67 + 2.30i)T + (-14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (0.378 + 0.274i)T + (16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (-5.09 + 1.65i)T + (47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (-1.96 - 0.638i)T + (49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + (1.25 - 0.908i)T + (20.7 - 63.7i)T^{2} \) |
| 71 | \( 1 + (-7.81 - 5.67i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (8.54 + 2.77i)T + (59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (-4.49 - 3.26i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (10.7 - 7.80i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + (-0.992 + 3.05i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (-9.69 + 13.3i)T + (-29.9 - 92.2i)T^{2} \) |
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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
−13.31041410779610057695431767524, −10.93951366431456476900941377896, −10.44158034944168227911041018542, −9.711651223978046201098120695115, −8.849711428948312305710756335917, −7.62295992756188620477720270869, −7.14372798873652695968026273975, −5.28411664447885750783828502221, −3.82454007047194825811909625032, −2.49573974352353659881548730900,
1.68603457608091715584952315900, 2.48426118027699139905768742164, 4.11517178704897723207609754875, 6.31918018124067164736381744861, 7.57210655451732180118492128075, 8.535738194406748998217363538248, 9.022112971996983537958166380360, 9.654213513189402927609599215791, 11.55556740766724037689700345211, 12.35705522820258531943647474177
