L(s) = 1 | + (0.642 + 0.766i)2-s + (0.613 − 1.68i)3-s + (−0.173 + 0.984i)4-s + (1.68 − 0.613i)6-s + (1.17 + 0.680i)7-s + (−0.866 + 0.500i)8-s + (−0.169 − 0.142i)9-s + (−3.22 − 5.59i)11-s + (1.55 + 0.897i)12-s + (−2.01 − 5.52i)13-s + (0.236 + 1.34i)14-s + (−0.939 − 0.342i)16-s + (1.64 + 1.96i)17-s − 0.221i·18-s + (3.83 − 2.06i)19-s + ⋯ |
L(s) = 1 | + (0.454 + 0.541i)2-s + (0.354 − 0.973i)3-s + (−0.0868 + 0.492i)4-s + (0.688 − 0.250i)6-s + (0.445 + 0.257i)7-s + (−0.306 + 0.176i)8-s + (−0.0566 − 0.0474i)9-s + (−0.973 − 1.68i)11-s + (0.448 + 0.259i)12-s + (−0.557 − 1.53i)13-s + (0.0631 + 0.358i)14-s + (−0.234 − 0.0855i)16-s + (0.398 + 0.475i)17-s − 0.0522i·18-s + (0.880 − 0.473i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.694 + 0.719i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.694 + 0.719i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.04604 - 0.868912i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.04604 - 0.868912i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.642 - 0.766i)T \) |
| 5 | \( 1 \) |
| 19 | \( 1 + (-3.83 + 2.06i)T \) |
good | 3 | \( 1 + (-0.613 + 1.68i)T + (-2.29 - 1.92i)T^{2} \) |
| 7 | \( 1 + (-1.17 - 0.680i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (3.22 + 5.59i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (2.01 + 5.52i)T + (-9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (-1.64 - 1.96i)T + (-2.95 + 16.7i)T^{2} \) |
| 23 | \( 1 + (-8.29 - 1.46i)T + (21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (3.41 + 2.86i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (-0.701 + 1.21i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 4.42iT - 37T^{2} \) |
| 41 | \( 1 + (-5.15 - 1.87i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (6.74 - 1.19i)T + (40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (6.11 - 7.29i)T + (-8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + (-4.35 - 0.768i)T + (49.8 + 18.1i)T^{2} \) |
| 59 | \( 1 + (-7.87 + 6.60i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (-1.12 + 6.36i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (4.31 - 5.14i)T + (-11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-0.336 - 1.90i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (2.14 - 5.90i)T + (-55.9 - 46.9i)T^{2} \) |
| 79 | \( 1 + (3.37 + 1.22i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (-11.6 - 6.74i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (1.78 - 0.648i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (-0.0793 - 0.0945i)T + (-16.8 + 95.5i)T^{2} \) |
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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
−9.896472975489673891730444326560, −8.626983695696149815912932782216, −7.974576116045383955320363041525, −7.63312277179822715429350853220, −6.55348924550344951353434239180, −5.48859675337631451430161220147, −5.07826424039939161863371704019, −3.27726156829097808110730009203, −2.68642436437849160367885509161, −0.919631574358283818357252510001,
1.68788064925699622874907344093, 2.84986281552792444554264880699, 3.98503702085555969669567130217, 4.79349021019831934101924036076, 5.18701885380160823263569081092, 6.96671578817536989855873616049, 7.41511899293814268241260389938, 8.890392872471904044747736907607, 9.548225087357564398057006892647, 10.08638263261690238876378621617