| L(s) = 1 | + (−1.21 − 2.38i)2-s + (−1.28 + 1.16i)3-s + (−3.02 + 4.16i)4-s + (0.0992 + 0.626i)5-s + (4.32 + 1.64i)6-s + (1.16 + 1.36i)7-s + (8.31 + 1.31i)8-s + (0.292 − 2.98i)9-s + (1.37 − 0.997i)10-s + (−1.80 + 2.94i)11-s + (−0.963 − 8.86i)12-s + (0.481 + 6.11i)13-s + (1.83 − 4.42i)14-s + (−0.856 − 0.688i)15-s + (−3.77 − 11.6i)16-s + (−0.501 − 2.08i)17-s + ⋯ |
| L(s) = 1 | + (−0.858 − 1.68i)2-s + (−0.740 + 0.671i)3-s + (−1.51 + 2.08i)4-s + (0.0444 + 0.280i)5-s + (1.76 + 0.671i)6-s + (0.440 + 0.515i)7-s + (2.94 + 0.465i)8-s + (0.0975 − 0.995i)9-s + (0.434 − 0.315i)10-s + (−0.544 + 0.888i)11-s + (−0.278 − 2.55i)12-s + (0.133 + 1.69i)13-s + (0.490 − 1.18i)14-s + (−0.221 − 0.177i)15-s + (−0.943 − 2.90i)16-s + (−0.121 − 0.506i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 123 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.958 - 0.285i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 123 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.958 - 0.285i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.446941 + 0.0652213i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.446941 + 0.0652213i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.28 - 1.16i)T \) |
| 41 | \( 1 + (3.45 + 5.39i)T \) |
| good | 2 | \( 1 + (1.21 + 2.38i)T + (-1.17 + 1.61i)T^{2} \) |
| 5 | \( 1 + (-0.0992 - 0.626i)T + (-4.75 + 1.54i)T^{2} \) |
| 7 | \( 1 + (-1.16 - 1.36i)T + (-1.09 + 6.91i)T^{2} \) |
| 11 | \( 1 + (1.80 - 2.94i)T + (-4.99 - 9.80i)T^{2} \) |
| 13 | \( 1 + (-0.481 - 6.11i)T + (-12.8 + 2.03i)T^{2} \) |
| 17 | \( 1 + (0.501 + 2.08i)T + (-15.1 + 7.71i)T^{2} \) |
| 19 | \( 1 + (-0.0229 + 0.291i)T + (-18.7 - 2.97i)T^{2} \) |
| 23 | \( 1 + (2.60 - 8.02i)T + (-18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (0.356 - 1.48i)T + (-25.8 - 13.1i)T^{2} \) |
| 31 | \( 1 + (2.28 + 3.13i)T + (-9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (2.56 + 1.86i)T + (11.4 + 35.1i)T^{2} \) |
| 43 | \( 1 + (2.08 - 1.06i)T + (25.2 - 34.7i)T^{2} \) |
| 47 | \( 1 + (-1.05 - 0.901i)T + (7.35 + 46.4i)T^{2} \) |
| 53 | \( 1 + (-2.83 - 0.679i)T + (47.2 + 24.0i)T^{2} \) |
| 59 | \( 1 + (-9.33 - 3.03i)T + (47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (-3.43 + 6.74i)T + (-35.8 - 49.3i)T^{2} \) |
| 67 | \( 1 + (-4.14 - 6.76i)T + (-30.4 + 59.6i)T^{2} \) |
| 71 | \( 1 + (-4.29 - 2.63i)T + (32.2 + 63.2i)T^{2} \) |
| 73 | \( 1 + (-2.75 - 2.75i)T + 73iT^{2} \) |
| 79 | \( 1 + (0.787 + 1.90i)T + (-55.8 + 55.8i)T^{2} \) |
| 83 | \( 1 + 13.0iT - 83T^{2} \) |
| 89 | \( 1 + (-1.38 + 1.18i)T + (13.9 - 87.9i)T^{2} \) |
| 97 | \( 1 + (-5.88 + 3.60i)T + (44.0 - 86.4i)T^{2} \) |
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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
−12.93968128158753201401966571776, −11.75036312789350801453120555521, −11.54914267358191246499523418501, −10.41095728302840984698857867193, −9.573932956868602019443698635925, −8.813832158360806601361913335740, −7.17910829749725767690587104081, −5.02665332219521603057822910566, −3.82453616505851065922855703168, −2.03461451827051435126432122987,
0.73010401629944433301965602889, 4.96209668450920593259416031567, 5.80917019471116334814976922872, 6.84407949145762020930348486584, 8.059226754378959171775106635751, 8.409904077067006375941004711572, 10.29111785705891349880391463687, 10.81097509728599564568563008853, 12.72230125619012511177243803537, 13.56115591155719002433046830597
