L(s) = 1 | + (−0.540 − 0.540i)2-s − 1.41i·4-s + (1.03 − 1.98i)5-s + (2.57 + 0.614i)7-s + (−1.84 + 1.84i)8-s + (−1.63 + 0.510i)10-s + 3.85·11-s + (−3.66 − 3.66i)13-s + (−1.05 − 1.72i)14-s − 0.839·16-s + (−1.49 + 1.49i)17-s + 0.0697·19-s + (−2.80 − 1.46i)20-s + (−2.08 − 2.08i)22-s + (0.534 − 0.534i)23-s + ⋯ |
L(s) = 1 | + (−0.381 − 0.381i)2-s − 0.708i·4-s + (0.463 − 0.886i)5-s + (0.972 + 0.232i)7-s + (−0.652 + 0.652i)8-s + (−0.515 + 0.161i)10-s + 1.16·11-s + (−1.01 − 1.01i)13-s + (−0.282 − 0.460i)14-s − 0.209·16-s + (−0.361 + 0.361i)17-s + 0.0160·19-s + (−0.627 − 0.328i)20-s + (−0.443 − 0.443i)22-s + (0.111 − 0.111i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.161 + 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.161 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.783623 - 0.922681i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.783623 - 0.922681i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (-1.03 + 1.98i)T \) |
| 7 | \( 1 + (-2.57 - 0.614i)T \) |
good | 2 | \( 1 + (0.540 + 0.540i)T + 2iT^{2} \) |
| 11 | \( 1 - 3.85T + 11T^{2} \) |
| 13 | \( 1 + (3.66 + 3.66i)T + 13iT^{2} \) |
| 17 | \( 1 + (1.49 - 1.49i)T - 17iT^{2} \) |
| 19 | \( 1 - 0.0697T + 19T^{2} \) |
| 23 | \( 1 + (-0.534 + 0.534i)T - 23iT^{2} \) |
| 29 | \( 1 + 2.77iT - 29T^{2} \) |
| 31 | \( 1 + 2.39iT - 31T^{2} \) |
| 37 | \( 1 + (-6.18 - 6.18i)T + 37iT^{2} \) |
| 41 | \( 1 + 8.68iT - 41T^{2} \) |
| 43 | \( 1 + (2.77 - 2.77i)T - 43iT^{2} \) |
| 47 | \( 1 + (5.49 - 5.49i)T - 47iT^{2} \) |
| 53 | \( 1 + (6.13 - 6.13i)T - 53iT^{2} \) |
| 59 | \( 1 - 6.97T + 59T^{2} \) |
| 61 | \( 1 - 14.3iT - 61T^{2} \) |
| 67 | \( 1 + (-0.416 - 0.416i)T + 67iT^{2} \) |
| 71 | \( 1 - 8.12T + 71T^{2} \) |
| 73 | \( 1 + (-9.55 - 9.55i)T + 73iT^{2} \) |
| 79 | \( 1 + 9.86iT - 79T^{2} \) |
| 83 | \( 1 + (1.63 + 1.63i)T + 83iT^{2} \) |
| 89 | \( 1 - 5.05T + 89T^{2} \) |
| 97 | \( 1 + (-6.85 + 6.85i)T - 97iT^{2} \) |
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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
−11.43094238991798112679082273637, −10.35009760832894304527385800864, −9.548915466544013831636491071955, −8.790834699668182587489920486606, −7.88156775457372118873862051941, −6.28538450906122068443181371140, −5.34716434369297877943105767339, −4.47319363479214679763325679583, −2.31398268915615705517266093093, −1.10087557538823301430177186745,
2.08911250615361279838011224130, 3.60572319243355704100439700382, 4.82489325763853502464693518480, 6.52865685182688344684262763782, 7.03566684060290238877344389703, 8.008317319184307878134581920755, 9.140140485181406409271641826538, 9.787444090206641399413435764719, 11.23378600208580871508519252502, 11.64276441478461656469613816210