| L(s) = 1 | + (−0.766 + 0.642i)2-s + (−1.68 − 0.613i)3-s + (0.173 − 0.984i)4-s + (0.173 + 0.984i)5-s + (1.68 − 0.613i)6-s + (−0.680 + 1.17i)7-s + (0.500 + 0.866i)8-s + (0.169 + 0.142i)9-s + (−0.766 − 0.642i)10-s + (−3.22 − 5.59i)11-s + (−0.897 + 1.55i)12-s + (−5.52 + 2.01i)13-s + (−0.236 − 1.34i)14-s + (0.311 − 1.76i)15-s + (−0.939 − 0.342i)16-s + (−1.96 + 1.64i)17-s + ⋯ |
| L(s) = 1 | + (−0.541 + 0.454i)2-s + (−0.973 − 0.354i)3-s + (0.0868 − 0.492i)4-s + (0.0776 + 0.440i)5-s + (0.688 − 0.250i)6-s + (−0.257 + 0.445i)7-s + (0.176 + 0.306i)8-s + (0.0566 + 0.0474i)9-s + (−0.242 − 0.203i)10-s + (−0.973 − 1.68i)11-s + (−0.259 + 0.448i)12-s + (−1.53 + 0.557i)13-s + (−0.0631 − 0.358i)14-s + (0.0804 − 0.456i)15-s + (−0.234 − 0.0855i)16-s + (−0.475 + 0.398i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 190 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.887 + 0.460i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 190 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.887 + 0.460i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0138879 - 0.0569378i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0138879 - 0.0569378i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.766 - 0.642i)T \) |
| 5 | \( 1 + (-0.173 - 0.984i)T \) |
| 19 | \( 1 + (3.83 - 2.06i)T \) |
| good | 3 | \( 1 + (1.68 + 0.613i)T + (2.29 + 1.92i)T^{2} \) |
| 7 | \( 1 + (0.680 - 1.17i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (3.22 + 5.59i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (5.52 - 2.01i)T + (9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (1.96 - 1.64i)T + (2.95 - 16.7i)T^{2} \) |
| 23 | \( 1 + (-1.46 + 8.29i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (-3.41 - 2.86i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (-0.701 + 1.21i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 4.42T + 37T^{2} \) |
| 41 | \( 1 + (-5.15 - 1.87i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (-1.19 - 6.74i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (7.29 + 6.11i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + (-0.768 + 4.35i)T + (-49.8 - 18.1i)T^{2} \) |
| 59 | \( 1 + (7.87 - 6.60i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (-1.12 + 6.36i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (5.14 + 4.31i)T + (11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (-0.336 - 1.90i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (-5.90 - 2.14i)T + (55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (-3.37 - 1.22i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (-6.74 + 11.6i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-1.78 + 0.648i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (0.0945 - 0.0793i)T + (16.8 - 95.5i)T^{2} \) |
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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
−12.05977104587789440703278426838, −10.96636692033103173923877156371, −10.41422298142556033754654290522, −9.004283530469060126518668480362, −8.059986776735140634389607259031, −6.65887098325696669602905758677, −6.11733030965945305171884621825, −4.97451974634307921206844321834, −2.66691206729761745339915015627, −0.06374269368462409012132998672,
2.41477678417114575671769099627, 4.53931352452197599360557075049, 5.26177611459537978671360037812, 6.98710609765761069970388805330, 7.83038473045830132935656843684, 9.410462907433512241481625796284, 10.11296354089439109917572022907, 10.82192481827822368849542550886, 12.00325665475792607268132549633, 12.59585097269332206425490979262
