L(s) = 1 | + (−0.775 − 2.72i)2-s + (−2.65 − 4.46i)3-s + (−6.79 + 4.21i)4-s + (5.27 − 5.27i)5-s + (−10.0 + 10.6i)6-s − 22.9·7-s + (16.7 + 15.2i)8-s + (−12.9 + 23.7i)9-s + (−18.4 − 10.2i)10-s + (−10.6 − 10.6i)11-s + (36.8 + 19.1i)12-s + (12.7 − 12.7i)13-s + (17.7 + 62.3i)14-s + (−37.5 − 9.57i)15-s + (28.3 − 57.3i)16-s − 134. i·17-s + ⋯ |
L(s) = 1 | + (−0.274 − 0.961i)2-s + (−0.510 − 0.859i)3-s + (−0.849 + 0.527i)4-s + (0.472 − 0.472i)5-s + (−0.686 + 0.726i)6-s − 1.23·7-s + (0.740 + 0.672i)8-s + (−0.478 + 0.878i)9-s + (−0.583 − 0.324i)10-s + (−0.291 − 0.291i)11-s + (0.887 + 0.461i)12-s + (0.271 − 0.271i)13-s + (0.339 + 1.19i)14-s + (−0.646 − 0.164i)15-s + (0.443 − 0.896i)16-s − 1.91i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.950 - 0.311i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.950 - 0.311i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.0979102 + 0.613534i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0979102 + 0.613534i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.775 + 2.72i)T \) |
| 3 | \( 1 + (2.65 + 4.46i)T \) |
good | 5 | \( 1 + (-5.27 + 5.27i)T - 125iT^{2} \) |
| 7 | \( 1 + 22.9T + 343T^{2} \) |
| 11 | \( 1 + (10.6 + 10.6i)T + 1.33e3iT^{2} \) |
| 13 | \( 1 + (-12.7 + 12.7i)T - 2.19e3iT^{2} \) |
| 17 | \( 1 + 134. iT - 4.91e3T^{2} \) |
| 19 | \( 1 + (46.9 + 46.9i)T + 6.85e3iT^{2} \) |
| 23 | \( 1 + 93.7iT - 1.21e4T^{2} \) |
| 29 | \( 1 + (-161. - 161. i)T + 2.43e4iT^{2} \) |
| 31 | \( 1 + 120. iT - 2.97e4T^{2} \) |
| 37 | \( 1 + (-2.42 - 2.42i)T + 5.06e4iT^{2} \) |
| 41 | \( 1 + 253.T + 6.89e4T^{2} \) |
| 43 | \( 1 + (135. - 135. i)T - 7.95e4iT^{2} \) |
| 47 | \( 1 - 468.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (-321. + 321. i)T - 1.48e5iT^{2} \) |
| 59 | \( 1 + (119. + 119. i)T + 2.05e5iT^{2} \) |
| 61 | \( 1 + (-310. + 310. i)T - 2.26e5iT^{2} \) |
| 67 | \( 1 + (-705. - 705. i)T + 3.00e5iT^{2} \) |
| 71 | \( 1 + 501. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + 641. iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 1.23e3iT - 4.93e5T^{2} \) |
| 83 | \( 1 + (-87.3 + 87.3i)T - 5.71e5iT^{2} \) |
| 89 | \( 1 + 1.19e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.05e3T + 9.12e5T^{2} \) |
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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
−13.80470889140523670144211500742, −13.13506150214681103403288224668, −12.31775689519782787474564263102, −11.09777648560103638234770795217, −9.811894413511394838516607080079, −8.609553923766304521266714284964, −6.88911590145714419397310607935, −5.19721080892675146525149712002, −2.74534372722940205296616776696, −0.55528101737796334534504951778,
3.94317512520753635886314456149, 5.83824717278603508413803815597, 6.58906506809900313939343815046, 8.591919300156645414945730915509, 9.967650886786838334908882033807, 10.43175082850671817634607391534, 12.49876040885783449622756001925, 13.79898961875820076631080696810, 15.07087032983912924587893029700, 15.76536732161635804637725163228