| L(s) = 1 | + (−0.304 + 0.527i)2-s + (1.79 − 0.653i)3-s + (0.814 + 1.41i)4-s + (0.460 + 0.386i)5-s + (−0.202 + 1.14i)6-s + (−2.95 − 2.48i)7-s − 2.21·8-s + (0.495 − 0.415i)9-s + (−0.343 + 0.125i)10-s + (3.72 + 1.35i)11-s + (2.38 + 1.99i)12-s + (−2.58 − 2.17i)13-s + (2.20 − 0.803i)14-s + (1.07 + 0.392i)15-s + (−0.955 + 1.65i)16-s + (2.43 − 4.22i)17-s + ⋯ |
| L(s) = 1 | + (−0.215 + 0.373i)2-s + (1.03 − 0.377i)3-s + (0.407 + 0.705i)4-s + (0.205 + 0.172i)5-s + (−0.0824 + 0.467i)6-s + (−1.11 − 0.937i)7-s − 0.781·8-s + (0.165 − 0.138i)9-s + (−0.108 + 0.0395i)10-s + (1.12 + 0.408i)11-s + (0.687 + 0.577i)12-s + (−0.718 − 0.602i)13-s + (0.590 − 0.214i)14-s + (0.278 + 0.101i)15-s + (−0.238 + 0.413i)16-s + (0.591 − 1.02i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 109 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.887 - 0.460i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 109 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.887 - 0.460i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.19668 + 0.291620i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.19668 + 0.291620i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 109 | \( 1 + (5.85 - 8.64i)T \) |
| good | 2 | \( 1 + (0.304 - 0.527i)T + (-1 - 1.73i)T^{2} \) |
| 3 | \( 1 + (-1.79 + 0.653i)T + (2.29 - 1.92i)T^{2} \) |
| 5 | \( 1 + (-0.460 - 0.386i)T + (0.868 + 4.92i)T^{2} \) |
| 7 | \( 1 + (2.95 + 2.48i)T + (1.21 + 6.89i)T^{2} \) |
| 11 | \( 1 + (-3.72 - 1.35i)T + (8.42 + 7.07i)T^{2} \) |
| 13 | \( 1 + (2.58 + 2.17i)T + (2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-2.43 + 4.22i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (2.93 - 5.07i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.40 + 2.42i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.37 + 0.499i)T + (22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (-1.06 + 0.895i)T + (5.38 - 30.5i)T^{2} \) |
| 37 | \( 1 + (6.69 - 5.61i)T + (6.42 - 36.4i)T^{2} \) |
| 41 | \( 1 - 4.49T + 41T^{2} \) |
| 43 | \( 1 + (-1.79 + 3.10i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-1.93 - 10.9i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + (-4.95 - 4.15i)T + (9.20 + 52.1i)T^{2} \) |
| 59 | \( 1 + (-5.30 - 1.93i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-1.90 + 10.7i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (4.90 - 4.11i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (1.19 + 2.06i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-13.9 - 5.09i)T + (55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (-2.69 + 2.26i)T + (13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (6.73 - 2.45i)T + (63.5 - 53.3i)T^{2} \) |
| 89 | \( 1 + (-1.07 + 6.09i)T + (-83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (-5.50 - 4.62i)T + (16.8 + 95.5i)T^{2} \) |
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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
−13.94748080837989553368898539704, −12.75692113814928908401259002908, −12.04625734214703673810057398773, −10.29276468102876820597737569608, −9.345478025155901267306159004056, −8.113434040434459909630693570051, −7.25323193460568407314982932024, −6.36427574870828544600631158270, −3.81958654570402606561257973060, −2.66381301341243409461882119702,
2.26474020083992832701888399082, 3.58345046840230482491476192569, 5.71354002602098248337215820369, 6.76426204339452920101954013594, 8.847634352919047993206826749804, 9.220973964256199210542567587476, 10.09986596009502095283199386150, 11.54100709018740123615666555569, 12.46181693899224869691648402864, 13.79189398574845011730319651294
