L(s) = 1 | + (1.64 − 0.951i)2-s + (−0.866 − 0.5i)3-s + (0.811 − 1.40i)4-s + (−1.76 + 1.37i)5-s − 1.90·6-s + 0.719i·8-s + (0.499 + 0.866i)9-s + (−1.59 + 3.94i)10-s + (−1 + 1.73i)11-s + (−1.40 + 0.811i)12-s + 6.42i·13-s + (2.21 − 0.311i)15-s + (2.30 + 3.99i)16-s + (−3.83 − 2.21i)17-s + (1.64 + 0.951i)18-s + (1.21 + 2.10i)19-s + ⋯ |
L(s) = 1 | + (1.16 − 0.672i)2-s + (−0.499 − 0.288i)3-s + (0.405 − 0.702i)4-s + (−0.788 + 0.615i)5-s − 0.776·6-s + 0.254i·8-s + (0.166 + 0.288i)9-s + (−0.504 + 1.24i)10-s + (−0.301 + 0.522i)11-s + (−0.405 + 0.234i)12-s + 1.78i·13-s + (0.571 − 0.0803i)15-s + (0.576 + 0.998i)16-s + (−0.930 − 0.537i)17-s + (0.388 + 0.224i)18-s + (0.278 + 0.482i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.595 - 0.803i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.595 - 0.803i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.40453 + 0.707528i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.40453 + 0.707528i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.866 + 0.5i)T \) |
| 5 | \( 1 + (1.76 - 1.37i)T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (-1.64 + 0.951i)T + (1 - 1.73i)T^{2} \) |
| 11 | \( 1 + (1 - 1.73i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 6.42iT - 13T^{2} \) |
| 17 | \( 1 + (3.83 + 2.21i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.21 - 2.10i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.19 + 0.688i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 0.755T + 29T^{2} \) |
| 31 | \( 1 + (-2.59 + 4.48i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (6.59 - 3.80i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 8.23T + 41T^{2} \) |
| 43 | \( 1 - 10.1iT - 43T^{2} \) |
| 47 | \( 1 + (-2.38 + 1.37i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (7.95 + 4.59i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (7.05 - 12.2i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-3.42 - 5.93i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.38 - 1.37i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 2T + 71T^{2} \) |
| 73 | \( 1 + (1.36 + 0.785i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (2.42 + 4.20i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 11.6iT - 83T^{2} \) |
| 89 | \( 1 + (2.31 + 4.00i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 11.9iT - 97T^{2} \) |
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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
−11.04294719722668126179361982477, −9.994823160405474130025506239797, −8.814077850227024371531587606623, −7.65064312983060622989570824881, −6.84119587114942922473957467019, −5.99699641986880919575526845151, −4.60180605541853840442684416198, −4.30597621853030100236148178461, −2.98584515500562604811386691058, −1.92401266069511684911111986296,
0.58790996674027425526039063301, 3.15490486241868750903995000822, 3.99544074375808703558698049769, 5.04653237913726654050203574952, 5.46059971382305711476970495336, 6.49287337326396511095689386456, 7.48522752893101142339456266769, 8.299319602354425567878437859626, 9.314706257873470478759097595544, 10.56333860038004172397518017285