L(s) = 1 | + (1.32 − 2.69i)3-s + 0.640·5-s + 2.72·7-s + (−5.47 − 7.14i)9-s − 11.2·11-s − 5.25i·13-s + (0.849 − 1.72i)15-s + 14.8i·17-s − 15.0i·19-s + (3.61 − 7.31i)21-s − 36.4i·23-s − 24.5·25-s + (−26.4 + 5.24i)27-s − 51.7·29-s + 36.5·31-s + ⋯ |
L(s) = 1 | + (0.442 − 0.896i)3-s + 0.128·5-s + 0.388·7-s + (−0.608 − 0.793i)9-s − 1.02·11-s − 0.403i·13-s + (0.0566 − 0.114i)15-s + 0.874i·17-s − 0.793i·19-s + (0.171 − 0.348i)21-s − 1.58i·23-s − 0.983·25-s + (−0.980 + 0.194i)27-s − 1.78·29-s + 1.17·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.947 + 0.321i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.947 + 0.321i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.250361454\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.250361454\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.32 + 2.69i)T \) |
good | 5 | \( 1 - 0.640T + 25T^{2} \) |
| 7 | \( 1 - 2.72T + 49T^{2} \) |
| 11 | \( 1 + 11.2T + 121T^{2} \) |
| 13 | \( 1 + 5.25iT - 169T^{2} \) |
| 17 | \( 1 - 14.8iT - 289T^{2} \) |
| 19 | \( 1 + 15.0iT - 361T^{2} \) |
| 23 | \( 1 + 36.4iT - 529T^{2} \) |
| 29 | \( 1 + 51.7T + 841T^{2} \) |
| 31 | \( 1 - 36.5T + 961T^{2} \) |
| 37 | \( 1 + 63.6iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 12.1iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 11.8iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 61.1iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 59.1T + 2.80e3T^{2} \) |
| 59 | \( 1 - 37.2T + 3.48e3T^{2} \) |
| 61 | \( 1 + 58.1iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 23.0iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 7.29iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 73.4T + 5.32e3T^{2} \) |
| 79 | \( 1 + 58.5T + 6.24e3T^{2} \) |
| 83 | \( 1 + 32.3T + 6.88e3T^{2} \) |
| 89 | \( 1 - 112. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 80.0T + 9.40e3T^{2} \) |
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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
−9.662458201395618471537520559319, −8.650680044280208348762409758866, −8.004874488853788801638838108560, −7.31156988377638831282238284042, −6.23958302755279795687161056612, −5.43743290229590807534904428604, −4.13793952846476284328540246471, −2.80299156425442718997590814885, −1.93449394414153898459687032933, −0.37397394063188883488233398943,
1.88285627471135063262706611869, 3.06869072360125078482897343678, 4.08546040981677585561333075510, 5.14019023676220657529178979087, 5.75828932635892989698560685867, 7.31847426184419936055581295230, 7.995398241647193357298597601120, 8.856211210965023724438459952221, 9.874435417756520400943130483604, 10.14739308739182190901896262101