L(s) = 1 | + (0.923 − 0.382i)2-s + (−2.33 − 1.56i)3-s + (0.707 − 0.707i)4-s + (−2.75 − 0.548i)6-s + (1.27 + 0.254i)7-s + (0.382 − 0.923i)8-s + (1.88 + 4.54i)9-s + (1.14 − 5.73i)11-s + (−2.75 + 0.548i)12-s + 5.33·13-s + (1.27 − 0.254i)14-s − i·16-s + (−3.93 + 1.22i)17-s + (3.47 + 3.47i)18-s + (0.311 − 0.752i)19-s + ⋯ |
L(s) = 1 | + (0.653 − 0.270i)2-s + (−1.35 − 0.902i)3-s + (0.353 − 0.353i)4-s + (−1.12 − 0.224i)6-s + (0.483 + 0.0961i)7-s + (0.135 − 0.326i)8-s + (0.626 + 1.51i)9-s + (0.343 − 1.72i)11-s + (−0.796 + 0.158i)12-s + 1.48·13-s + (0.341 − 0.0679i)14-s − 0.250i·16-s + (−0.954 + 0.296i)17-s + (0.819 + 0.819i)18-s + (0.0715 − 0.172i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.784 + 0.620i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.784 + 0.620i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.461107 - 1.32712i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.461107 - 1.32712i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.923 + 0.382i)T \) |
| 5 | \( 1 \) |
| 17 | \( 1 + (3.93 - 1.22i)T \) |
good | 3 | \( 1 + (2.33 + 1.56i)T + (1.14 + 2.77i)T^{2} \) |
| 7 | \( 1 + (-1.27 - 0.254i)T + (6.46 + 2.67i)T^{2} \) |
| 11 | \( 1 + (-1.14 + 5.73i)T + (-10.1 - 4.20i)T^{2} \) |
| 13 | \( 1 - 5.33T + 13T^{2} \) |
| 19 | \( 1 + (-0.311 + 0.752i)T + (-13.4 - 13.4i)T^{2} \) |
| 23 | \( 1 + (3.88 + 5.82i)T + (-8.80 + 21.2i)T^{2} \) |
| 29 | \( 1 + (-1.17 + 1.75i)T + (-11.0 - 26.7i)T^{2} \) |
| 31 | \( 1 + (-0.301 - 1.51i)T + (-28.6 + 11.8i)T^{2} \) |
| 37 | \( 1 + (3.87 - 5.79i)T + (-14.1 - 34.1i)T^{2} \) |
| 41 | \( 1 + (4.40 + 6.59i)T + (-15.6 + 37.8i)T^{2} \) |
| 43 | \( 1 + (-1.31 - 0.544i)T + (30.4 + 30.4i)T^{2} \) |
| 47 | \( 1 + 0.109iT - 47T^{2} \) |
| 53 | \( 1 + (0.379 + 0.915i)T + (-37.4 + 37.4i)T^{2} \) |
| 59 | \( 1 + (-5.08 + 2.10i)T + (41.7 - 41.7i)T^{2} \) |
| 61 | \( 1 + (4.82 - 3.22i)T + (23.3 - 56.3i)T^{2} \) |
| 67 | \( 1 + (4.12 + 4.12i)T + 67iT^{2} \) |
| 71 | \( 1 + (5.38 - 1.07i)T + (65.5 - 27.1i)T^{2} \) |
| 73 | \( 1 + (-8.02 + 1.59i)T + (67.4 - 27.9i)T^{2} \) |
| 79 | \( 1 + (-5.01 - 0.997i)T + (72.9 + 30.2i)T^{2} \) |
| 83 | \( 1 + (-2.38 + 0.987i)T + (58.6 - 58.6i)T^{2} \) |
| 89 | \( 1 + (0.779 - 0.779i)T - 89iT^{2} \) |
| 97 | \( 1 + (-14.7 + 2.93i)T + (89.6 - 37.1i)T^{2} \) |
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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
−10.50040231719546433617831784185, −8.753683866009343905987241288389, −8.179510762470275935773844215601, −6.74587585640424822739366155777, −6.25020818706313403379382493123, −5.66937640338043905619396161053, −4.61428463336351685673100562089, −3.45050370207269955436197758248, −1.83833411021270619985586838340, −0.68420269504690702962921505124,
1.72175171731884765266356722469, 3.71886305511748481650995075248, 4.42488698370144637345132545777, 5.10589250391248193451858079883, 6.02949955301165835519596020892, 6.74087453480085347421114530207, 7.72770093937071781498848972680, 9.008469457385841061216875078742, 9.894076743874375920395858728091, 10.69143747156640147153696243880