| L(s) = 1 | + (−0.582 − 0.694i)2-s + (−1.06 + 1.36i)3-s + (0.204 − 1.16i)4-s + (−0.931 − 0.781i)5-s + (1.56 − 0.0508i)6-s + (−2.06 + 1.65i)7-s + (−2.49 + 1.44i)8-s + (−0.711 − 2.91i)9-s + 1.10i·10-s + (−2.30 − 2.74i)11-s + (1.36 + 1.52i)12-s + (0.206 + 0.566i)13-s + (2.35 + 0.473i)14-s + (2.06 − 0.432i)15-s + (0.238 + 0.0867i)16-s − 7.94·17-s + ⋯ |
| L(s) = 1 | + (−0.411 − 0.490i)2-s + (−0.617 + 0.786i)3-s + (0.102 − 0.580i)4-s + (−0.416 − 0.349i)5-s + (0.640 − 0.0207i)6-s + (−0.781 + 0.623i)7-s + (−0.882 + 0.509i)8-s + (−0.237 − 0.971i)9-s + 0.348i·10-s + (−0.695 − 0.828i)11-s + (0.393 + 0.438i)12-s + (0.0571 + 0.157i)13-s + (0.628 + 0.126i)14-s + (0.532 − 0.111i)15-s + (0.0595 + 0.0216i)16-s − 1.92·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.00433i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.00433i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.000341516 - 0.157391i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.000341516 - 0.157391i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.06 - 1.36i)T \) |
| 7 | \( 1 + (2.06 - 1.65i)T \) |
| good | 2 | \( 1 + (0.582 + 0.694i)T + (-0.347 + 1.96i)T^{2} \) |
| 5 | \( 1 + (0.931 + 0.781i)T + (0.868 + 4.92i)T^{2} \) |
| 11 | \( 1 + (2.30 + 2.74i)T + (-1.91 + 10.8i)T^{2} \) |
| 13 | \( 1 + (-0.206 - 0.566i)T + (-9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + 7.94T + 17T^{2} \) |
| 19 | \( 1 - 1.41iT - 19T^{2} \) |
| 23 | \( 1 + (0.349 + 0.958i)T + (-17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (0.413 - 1.13i)T + (-22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (-8.17 - 1.44i)T + (29.1 + 10.6i)T^{2} \) |
| 37 | \( 1 + (4.22 + 7.31i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-6.73 + 2.45i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (-1.41 - 8.02i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (0.0548 + 0.310i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + (-5.74 + 3.31i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (0.936 - 0.341i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (12.0 - 2.12i)T + (57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (7.90 + 6.63i)T + (11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (0.952 + 0.550i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (0.808 + 0.466i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (10.3 - 8.64i)T + (13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (5.66 + 2.06i)T + (63.5 + 53.3i)T^{2} \) |
| 89 | \( 1 + 5.92T + 89T^{2} \) |
| 97 | \( 1 + (8.29 - 1.46i)T + (91.1 - 33.1i)T^{2} \) |
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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
−11.78882439461567010173904071538, −10.97656705790320757273678810768, −10.25841714298013184356578877039, −9.178764031294807933477389879800, −8.574493572032051134388155945469, −6.47904641669316494548548235803, −5.70989739296021327974815113422, −4.42649331075950972822613175276, −2.75211182364333206936265779861, −0.15924241250387371862464249774,
2.71819848077755877626570606261, 4.42008144390677582257568475772, 6.23287796058974606390401390037, 7.05646862787038785905537388012, 7.60161540983296828374446492400, 8.800185223347780408375311873602, 10.16927956139012375288791848253, 11.20329838737593131919656290445, 12.12659912220424025350049158347, 13.08717340188065010710037028680
