L(s) = 1 | + (−0.691 − 1.23i)2-s + (−1.04 + 1.70i)4-s + (0.766 + 0.442i)5-s + (−2.19 − 1.48i)7-s + (2.82 + 0.108i)8-s + (0.0160 − 1.25i)10-s + (1.90 − 1.10i)11-s + 0.687i·13-s + (−0.313 + 3.72i)14-s + (−1.81 − 3.56i)16-s + (0.0941 − 0.0543i)17-s + (0.801 − 1.38i)19-s + (−1.55 + 0.845i)20-s + (−2.68 − 1.59i)22-s + (6.16 + 3.56i)23-s + ⋯ |
L(s) = 1 | + (−0.488 − 0.872i)2-s + (−0.522 + 0.852i)4-s + (0.342 + 0.197i)5-s + (−0.828 − 0.560i)7-s + (0.999 + 0.0385i)8-s + (0.00508 − 0.395i)10-s + (0.575 − 0.332i)11-s + 0.190i·13-s + (−0.0838 + 0.996i)14-s + (−0.454 − 0.890i)16-s + (0.0228 − 0.0131i)17-s + (0.183 − 0.318i)19-s + (−0.347 + 0.189i)20-s + (−0.571 − 0.339i)22-s + (1.28 + 0.742i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.221 + 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.221 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.649013 - 0.813115i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.649013 - 0.813115i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.691 + 1.23i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (2.19 + 1.48i)T \) |
good | 5 | \( 1 + (-0.766 - 0.442i)T + (2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-1.90 + 1.10i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 0.687iT - 13T^{2} \) |
| 17 | \( 1 + (-0.0941 + 0.0543i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.801 + 1.38i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-6.16 - 3.56i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 3.82T + 29T^{2} \) |
| 31 | \( 1 + (3.76 + 6.51i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-4.04 + 7.00i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 8.74iT - 41T^{2} \) |
| 43 | \( 1 - 2.09iT - 43T^{2} \) |
| 47 | \( 1 + (-6.62 + 11.4i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (3.78 + 6.54i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-3.23 - 5.60i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.33 - 0.772i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (3.46 - 1.99i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 0.352iT - 71T^{2} \) |
| 73 | \( 1 + (-0.816 + 0.471i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (4.08 + 2.35i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 6.80T + 83T^{2} \) |
| 89 | \( 1 + (11.7 + 6.76i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 10.8iT - 97T^{2} \) |
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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
−10.08408279353271356858441151189, −9.377105490395810432181363527338, −8.752726083482446086922272994204, −7.50729333354378392740665268301, −6.82278471455468814964134856936, −5.63392016452152573370200489445, −4.21841480196920841386841575572, −3.42199788474560478787745174778, −2.28159327254496380898208237769, −0.71516497867640235824376753026,
1.31190520074115376466683393894, 3.00016671903147898925968678053, 4.50274193453432275574792616972, 5.46094508308919925906143191234, 6.33568550808828923926075575995, 6.97354080858233578196304639516, 8.059979794112069452159277724588, 9.006587907836634071640026850323, 9.464061838599112809314762354202, 10.26052692252141745367924005144