L(s) = 1 | + (0.281 + 1.04i)2-s + (0.710 − 0.410i)4-s + (1.02 + 1.02i)5-s + (2.23 − 1.41i)7-s + (2.16 + 2.16i)8-s + (−0.788 + 1.36i)10-s + (−0.972 + 0.260i)11-s + (−3.05 + 1.91i)13-s + (2.11 + 1.94i)14-s + (−0.842 + 1.45i)16-s + (2.37 + 4.10i)17-s + (0.391 − 1.46i)19-s + (1.15 + 0.308i)20-s + (−0.546 − 0.946i)22-s + (0.337 + 0.194i)23-s + ⋯ |
L(s) = 1 | + (0.198 + 0.741i)2-s + (0.355 − 0.205i)4-s + (0.459 + 0.459i)5-s + (0.843 − 0.536i)7-s + (0.765 + 0.765i)8-s + (−0.249 + 0.432i)10-s + (−0.293 + 0.0785i)11-s + (−0.847 + 0.530i)13-s + (0.565 + 0.519i)14-s + (−0.210 + 0.364i)16-s + (0.574 + 0.995i)17-s + (0.0899 − 0.335i)19-s + (0.257 + 0.0690i)20-s + (−0.116 − 0.201i)22-s + (0.0702 + 0.0405i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.464 - 0.885i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.464 - 0.885i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.01539 + 1.21832i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.01539 + 1.21832i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (-2.23 + 1.41i)T \) |
| 13 | \( 1 + (3.05 - 1.91i)T \) |
good | 2 | \( 1 + (-0.281 - 1.04i)T + (-1.73 + i)T^{2} \) |
| 5 | \( 1 + (-1.02 - 1.02i)T + 5iT^{2} \) |
| 11 | \( 1 + (0.972 - 0.260i)T + (9.52 - 5.5i)T^{2} \) |
| 17 | \( 1 + (-2.37 - 4.10i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.391 + 1.46i)T + (-16.4 - 9.5i)T^{2} \) |
| 23 | \( 1 + (-0.337 - 0.194i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-4.30 + 7.44i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-2.92 - 2.92i)T + 31iT^{2} \) |
| 37 | \( 1 + (-9.77 + 2.61i)T + (32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + (8.64 - 2.31i)T + (35.5 - 20.5i)T^{2} \) |
| 43 | \( 1 + (8.28 - 4.78i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (4.78 - 4.78i)T - 47iT^{2} \) |
| 53 | \( 1 + 12.7T + 53T^{2} \) |
| 59 | \( 1 + (0.889 + 0.238i)T + (51.0 + 29.5i)T^{2} \) |
| 61 | \( 1 + (-8.36 + 4.82i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.71 - 6.39i)T + (-58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + (-2.56 - 0.687i)T + (61.4 + 35.5i)T^{2} \) |
| 73 | \( 1 + (2.46 - 2.46i)T - 73iT^{2} \) |
| 79 | \( 1 - 7.06T + 79T^{2} \) |
| 83 | \( 1 + (2.43 + 2.43i)T + 83iT^{2} \) |
| 89 | \( 1 + (2.79 + 10.4i)T + (-77.0 + 44.5i)T^{2} \) |
| 97 | \( 1 + (-3.55 + 13.2i)T + (-84.0 - 48.5i)T^{2} \) |
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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
−10.24551307333412605712505052371, −9.831749123476402752492328617201, −8.207817604127986207910413137024, −7.83763633848190341495175758441, −6.75450313224073124384731433675, −6.23176719264120881684191785690, −5.09285424020047099848013596300, −4.39786184331759399391481027049, −2.68999218415822877585442926290, −1.60178860101003038067762845533,
1.30166649201670704504626910018, 2.41834348085177606200865951777, 3.31756544870798058342280176314, 4.85693557200273752073983553186, 5.27873870878616236065094279301, 6.65483573047963071061866107343, 7.65921020163023852502723428364, 8.332244790049989266416939787075, 9.508402699191795438149782977308, 10.13352373391087824302564813654