| L(s) = 1 | + (−0.743 + 2.53i)2-s + (−1.34 − 1.16i)3-s + (−4.18 − 2.68i)4-s + (−2.23 − 0.00123i)5-s + (3.95 − 2.54i)6-s + (1.87 − 0.855i)7-s + (5.92 − 5.13i)8-s + (0.0244 + 0.170i)9-s + (1.66 − 5.66i)10-s + (−5.55 + 1.63i)11-s + (2.49 + 8.49i)12-s + (−2.87 − 1.31i)13-s + (0.773 + 5.38i)14-s + (3.00 + 2.60i)15-s + (4.47 + 9.78i)16-s + (−1.10 − 1.71i)17-s + ⋯ |
| L(s) = 1 | + (−0.525 + 1.79i)2-s + (−0.777 − 0.673i)3-s + (−2.09 − 1.34i)4-s + (−0.999 − 0.000551i)5-s + (1.61 − 1.03i)6-s + (0.707 − 0.323i)7-s + (2.09 − 1.81i)8-s + (0.00816 + 0.0567i)9-s + (0.526 − 1.79i)10-s + (−1.67 + 0.492i)11-s + (0.719 + 2.45i)12-s + (−0.796 − 0.363i)13-s + (0.206 + 1.43i)14-s + (0.776 + 0.673i)15-s + (1.11 + 2.44i)16-s + (−0.268 − 0.417i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.193 + 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.193 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0876069 - 0.0720051i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0876069 - 0.0720051i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (2.23 + 0.00123i)T \) |
| 23 | \( 1 + (1.21 - 4.63i)T \) |
| good | 2 | \( 1 + (0.743 - 2.53i)T + (-1.68 - 1.08i)T^{2} \) |
| 3 | \( 1 + (1.34 + 1.16i)T + (0.426 + 2.96i)T^{2} \) |
| 7 | \( 1 + (-1.87 + 0.855i)T + (4.58 - 5.29i)T^{2} \) |
| 11 | \( 1 + (5.55 - 1.63i)T + (9.25 - 5.94i)T^{2} \) |
| 13 | \( 1 + (2.87 + 1.31i)T + (8.51 + 9.82i)T^{2} \) |
| 17 | \( 1 + (1.10 + 1.71i)T + (-7.06 + 15.4i)T^{2} \) |
| 19 | \( 1 + (1.77 + 1.13i)T + (7.89 + 17.2i)T^{2} \) |
| 29 | \( 1 + (-1.74 + 1.11i)T + (12.0 - 26.3i)T^{2} \) |
| 31 | \( 1 + (1.04 + 1.20i)T + (-4.41 + 30.6i)T^{2} \) |
| 37 | \( 1 + (-0.620 + 0.0892i)T + (35.5 - 10.4i)T^{2} \) |
| 41 | \( 1 + (-0.289 + 2.01i)T + (-39.3 - 11.5i)T^{2} \) |
| 43 | \( 1 + (3.14 + 2.72i)T + (6.11 + 42.5i)T^{2} \) |
| 47 | \( 1 + 0.403iT - 47T^{2} \) |
| 53 | \( 1 + (-7.50 + 3.42i)T + (34.7 - 40.0i)T^{2} \) |
| 59 | \( 1 + (-3.73 + 8.17i)T + (-38.6 - 44.5i)T^{2} \) |
| 61 | \( 1 + (-0.891 - 1.02i)T + (-8.68 + 60.3i)T^{2} \) |
| 67 | \( 1 + (1.89 - 6.45i)T + (-56.3 - 36.2i)T^{2} \) |
| 71 | \( 1 + (13.3 + 3.91i)T + (59.7 + 38.3i)T^{2} \) |
| 73 | \( 1 + (-5.58 + 8.68i)T + (-30.3 - 66.4i)T^{2} \) |
| 79 | \( 1 + (3.54 - 7.75i)T + (-51.7 - 59.7i)T^{2} \) |
| 83 | \( 1 + (8.15 - 1.17i)T + (79.6 - 23.3i)T^{2} \) |
| 89 | \( 1 + (-2.09 + 2.41i)T + (-12.6 - 88.0i)T^{2} \) |
| 97 | \( 1 + (9.91 + 1.42i)T + (93.0 + 27.3i)T^{2} \) |
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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
−13.51879248476756916163549955383, −12.52141294445317124337091492983, −11.21558776737404421846646678056, −9.978591675433888089145277893109, −8.419575755283017350425305879279, −7.52411158713909097924575155587, −7.09424446550014921267751103190, −5.55721676861352435705218101184, −4.68915387490177524824104342953, −0.15794408878243766500998845186,
2.57841889750593929167831658968, 4.27364999019733754549977291471, 5.10970582675978164016465210962, 7.891582225113656302811180817372, 8.624104562878914363189616174618, 10.22212738891416907129349813658, 10.71817810944585650663355963574, 11.50011383172160284020597900163, 12.24781686903606831140592725863, 13.23827678872808760428836134539
