| L(s) = 1 | + (−0.781 + 0.623i)2-s + (1.57 − 0.618i)3-s + (0.222 − 0.974i)4-s + (1.91 − 1.14i)5-s + (−0.845 + 1.46i)6-s + (0.731 − 0.422i)7-s + (0.433 + 0.900i)8-s + (−0.101 + 0.0939i)9-s + (−0.782 + 2.09i)10-s + (1.27 + 5.60i)11-s + (−0.252 − 1.67i)12-s + (1.25 − 1.84i)13-s + (−0.308 + 0.786i)14-s + (2.30 − 2.99i)15-s + (−0.900 − 0.433i)16-s + (2.04 − 0.153i)17-s + ⋯ |
| L(s) = 1 | + (−0.552 + 0.440i)2-s + (0.909 − 0.356i)3-s + (0.111 − 0.487i)4-s + (0.857 − 0.514i)5-s + (−0.345 + 0.598i)6-s + (0.276 − 0.159i)7-s + (0.153 + 0.318i)8-s + (−0.0337 + 0.0313i)9-s + (−0.247 + 0.662i)10-s + (0.385 + 1.68i)11-s + (−0.0727 − 0.482i)12-s + (0.348 − 0.510i)13-s + (−0.0824 + 0.210i)14-s + (0.596 − 0.773i)15-s + (−0.225 − 0.108i)16-s + (0.497 − 0.0372i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00548i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.00548i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.66850 - 0.00457377i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.66850 - 0.00457377i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.781 - 0.623i)T \) |
| 5 | \( 1 + (-1.91 + 1.14i)T \) |
| 43 | \( 1 + (-6.35 - 1.61i)T \) |
| good | 3 | \( 1 + (-1.57 + 0.618i)T + (2.19 - 2.04i)T^{2} \) |
| 7 | \( 1 + (-0.731 + 0.422i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.27 - 5.60i)T + (-9.91 + 4.77i)T^{2} \) |
| 13 | \( 1 + (-1.25 + 1.84i)T + (-4.74 - 12.1i)T^{2} \) |
| 17 | \( 1 + (-2.04 + 0.153i)T + (16.8 - 2.53i)T^{2} \) |
| 19 | \( 1 + (1.51 + 1.40i)T + (1.41 + 18.9i)T^{2} \) |
| 23 | \( 1 + (-2.15 + 6.97i)T + (-19.0 - 12.9i)T^{2} \) |
| 29 | \( 1 + (-0.229 + 0.585i)T + (-21.2 - 19.7i)T^{2} \) |
| 31 | \( 1 + (4.82 - 0.726i)T + (29.6 - 9.13i)T^{2} \) |
| 37 | \( 1 + (-0.338 - 0.195i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (5.05 + 6.34i)T + (-9.12 + 39.9i)T^{2} \) |
| 47 | \( 1 + (6.26 + 1.43i)T + (42.3 + 20.3i)T^{2} \) |
| 53 | \( 1 + (-2.36 - 3.46i)T + (-19.3 + 49.3i)T^{2} \) |
| 59 | \( 1 + (-13.0 - 6.30i)T + (36.7 + 46.1i)T^{2} \) |
| 61 | \( 1 + (13.6 + 2.06i)T + (58.2 + 17.9i)T^{2} \) |
| 67 | \( 1 + (8.48 - 9.14i)T + (-5.00 - 66.8i)T^{2} \) |
| 71 | \( 1 + (-1.32 + 0.407i)T + (58.6 - 39.9i)T^{2} \) |
| 73 | \( 1 + (-1.37 + 2.01i)T + (-26.6 - 67.9i)T^{2} \) |
| 79 | \( 1 + (1.14 + 1.98i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (9.01 - 3.53i)T + (60.8 - 56.4i)T^{2} \) |
| 89 | \( 1 + (-1.98 - 5.05i)T + (-65.2 + 60.5i)T^{2} \) |
| 97 | \( 1 + (9.05 - 2.06i)T + (87.3 - 42.0i)T^{2} \) |
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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
−10.81272103814870915604305967408, −10.01735804473392063675029777367, −9.133761120489255904331418696653, −8.531958730766598100266365340061, −7.57466722742773286816002024017, −6.75352081432629781472086456491, −5.51083373857305923844712125880, −4.46019116341705030682509318746, −2.55912580019837821643050912686, −1.52676087189097214828502753602,
1.63526680772880045402179171381, 3.04317114080079726072102135162, 3.66888292967306592501054865120, 5.55835203250694395599414472816, 6.48441581898829610764845796132, 7.83710095844463700515243298625, 8.739288495886483332156533775506, 9.258881973904807218862157287620, 10.09203768166802659173706940941, 11.14880601266021376641638895429
