L(s) = 1 | + (0.258 + 0.965i)2-s + (−0.965 − 0.258i)3-s + (−0.866 + 0.499i)4-s + (−2.21 + 0.296i)5-s − i·6-s + (−1.87 + 1.86i)7-s + (−0.707 − 0.707i)8-s + (0.866 + 0.499i)9-s + (−0.860 − 2.06i)10-s + (−2.74 − 4.75i)11-s + (0.965 − 0.258i)12-s + (−2.41 + 2.41i)13-s + (−2.28 − 1.33i)14-s + (2.21 + 0.286i)15-s + (0.500 − 0.866i)16-s + (−0.548 + 2.04i)17-s + ⋯ |
L(s) = 1 | + (0.183 + 0.683i)2-s + (−0.557 − 0.149i)3-s + (−0.433 + 0.249i)4-s + (−0.991 + 0.132i)5-s − 0.408i·6-s + (−0.709 + 0.704i)7-s + (−0.249 − 0.249i)8-s + (0.288 + 0.166i)9-s + (−0.272 − 0.652i)10-s + (−0.827 − 1.43i)11-s + (0.278 − 0.0747i)12-s + (−0.670 + 0.670i)13-s + (−0.611 − 0.355i)14-s + (0.572 + 0.0740i)15-s + (0.125 − 0.216i)16-s + (−0.132 + 0.496i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.972 + 0.232i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.972 + 0.232i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0328125 - 0.278700i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0328125 - 0.278700i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.258 - 0.965i)T \) |
| 3 | \( 1 + (0.965 + 0.258i)T \) |
| 5 | \( 1 + (2.21 - 0.296i)T \) |
| 7 | \( 1 + (1.87 - 1.86i)T \) |
good | 11 | \( 1 + (2.74 + 4.75i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (2.41 - 2.41i)T - 13iT^{2} \) |
| 17 | \( 1 + (0.548 - 2.04i)T + (-14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (3.49 - 6.05i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.69 + 0.454i)T + (19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 + 0.684iT - 29T^{2} \) |
| 31 | \( 1 + (-4.82 + 2.78i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-2.53 - 9.46i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 - 2.50iT - 41T^{2} \) |
| 43 | \( 1 + (1.95 + 1.95i)T + 43iT^{2} \) |
| 47 | \( 1 + (3.40 - 0.912i)T + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (2.43 - 9.08i)T + (-45.8 - 26.5i)T^{2} \) |
| 59 | \( 1 + (5.08 + 8.79i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.01 - 0.585i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (9.61 + 2.57i)T + (58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + 11.9T + 71T^{2} \) |
| 73 | \( 1 + (-4.70 - 1.26i)T + (63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (-7.21 - 4.16i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (4.05 - 4.05i)T - 83iT^{2} \) |
| 89 | \( 1 + (3.59 - 6.22i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (13.1 + 13.1i)T + 97iT^{2} \) |
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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
−12.77044983265811661638634353653, −12.06887215345752848377416507072, −11.12886218248996157589705320408, −9.989660184804894886195247257862, −8.569633462695173900299504971745, −7.909588318937060328364389026538, −6.57033491024264016155187399592, −5.85536499091866613338858774614, −4.50885825426416710056657530088, −3.11204942711360329858358162761,
0.23162469866311737990408210640, 2.82375918523490761074377208946, 4.32542690307275920512639207640, 5.02094548734674292231612110782, 6.85519020053812598030555980323, 7.62892555407222454778862662086, 9.177646326591376917687427519327, 10.21828635282317829358136837790, 10.83257286345553837499839029018, 11.91640524750993738293377166332