L(s) = 1 | + (−0.141 − 1.99i)2-s + (2.06 − 2.17i)3-s + (−3.95 + 0.565i)4-s + (−3.07 − 3.94i)5-s + (−4.63 − 3.81i)6-s + (5.18 + 5.18i)7-s + (1.68 + 7.81i)8-s + (−0.459 − 8.98i)9-s + (−7.42 + 6.69i)10-s + 7.14·11-s + (−6.95 + 9.78i)12-s + (−7.93 − 7.93i)13-s + (9.61 − 11.0i)14-s + (−14.9 − 1.45i)15-s + (15.3 − 4.47i)16-s + (16.5 + 16.5i)17-s + ⋯ |
L(s) = 1 | + (−0.0708 − 0.997i)2-s + (0.688 − 0.724i)3-s + (−0.989 + 0.141i)4-s + (−0.615 − 0.788i)5-s + (−0.771 − 0.635i)6-s + (0.741 + 0.741i)7-s + (0.211 + 0.977i)8-s + (−0.0510 − 0.998i)9-s + (−0.742 + 0.669i)10-s + 0.649·11-s + (−0.579 + 0.815i)12-s + (−0.610 − 0.610i)13-s + (0.686 − 0.791i)14-s + (−0.995 − 0.0970i)15-s + (0.960 − 0.279i)16-s + (0.975 + 0.975i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.487 + 0.873i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.487 + 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.627132 - 1.06805i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.627132 - 1.06805i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.141 + 1.99i)T \) |
| 3 | \( 1 + (-2.06 + 2.17i)T \) |
| 5 | \( 1 + (3.07 + 3.94i)T \) |
good | 7 | \( 1 + (-5.18 - 5.18i)T + 49iT^{2} \) |
| 11 | \( 1 - 7.14T + 121T^{2} \) |
| 13 | \( 1 + (7.93 + 7.93i)T + 169iT^{2} \) |
| 17 | \( 1 + (-16.5 - 16.5i)T + 289iT^{2} \) |
| 19 | \( 1 - 12.1T + 361T^{2} \) |
| 23 | \( 1 + (-11.0 - 11.0i)T + 529iT^{2} \) |
| 29 | \( 1 + 26.1T + 841T^{2} \) |
| 31 | \( 1 - 8.74iT - 961T^{2} \) |
| 37 | \( 1 + (26.7 - 26.7i)T - 1.36e3iT^{2} \) |
| 41 | \( 1 + 35.4iT - 1.68e3T^{2} \) |
| 43 | \( 1 + (24.6 - 24.6i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (58.6 - 58.6i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + (-20.4 + 20.4i)T - 2.80e3iT^{2} \) |
| 59 | \( 1 + 59.4iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 7.42T + 3.72e3T^{2} \) |
| 67 | \( 1 + (-35.8 - 35.8i)T + 4.48e3iT^{2} \) |
| 71 | \( 1 + 46.2T + 5.04e3T^{2} \) |
| 73 | \( 1 + (-10.6 - 10.6i)T + 5.32e3iT^{2} \) |
| 79 | \( 1 - 68.3T + 6.24e3T^{2} \) |
| 83 | \( 1 + (-76.6 - 76.6i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 + 41.0T + 7.92e3T^{2} \) |
| 97 | \( 1 + (-81.7 + 81.7i)T - 9.40e3iT^{2} \) |
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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
−14.35435344268478036893999282016, −12.98920010558247729221380314942, −12.24322024973181737946501236716, −11.48853148277838538548343044870, −9.627350167327223396938052648253, −8.563648384889834435154105980298, −7.77442845090183699804159297294, −5.26310082951561988648122038188, −3.47352667329462063248270701246, −1.49424293576142110239934767359,
3.67595935969080502441104486883, 4.92573237933756719847504297073, 7.08622347256304446753216816322, 7.83923092605009359244435837325, 9.240135218792550567951502108255, 10.33374992423712664767335828921, 11.68372920558483051429109415017, 13.70565106353957756002580546016, 14.46220797159296828913254125055, 14.95171109182873527425516425568