| 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
−13.23827678872808760428836134539, −12.24781686903606831140592725863, −11.50011383172160284020597900163, −10.71817810944585650663355963574, −10.22212738891416907129349813658, −8.624104562878914363189616174618, −7.891582225113656302811180817372, −5.10970582675978164016465210962, −4.27364999019733754549977291471, −2.57841889750593929167831658968,
0.15794408878243766500998845186, 4.68915387490177524824104342953, 5.55721676861352435705218101184, 7.09424446550014921267751103190, 7.52411158713909097924575155587, 8.419575755283017350425305879279, 9.978591675433888089145277893109, 11.21558776737404421846646678056, 12.52141294445317124337091492983, 13.51879248476756916163549955383
