| L(s) = 1 | + (−0.790 − 0.942i)2-s + (0.0284 + 0.161i)3-s + (0.0846 − 0.480i)4-s + (0.850 − 0.150i)5-s + (0.129 − 0.154i)6-s + (−0.990 − 2.45i)7-s + (−2.64 + 1.52i)8-s + (2.79 − 1.01i)9-s + (−0.813 − 0.682i)10-s + 1.91·11-s + 0.0798·12-s + (−3.99 − 3.35i)13-s + (−1.52 + 2.87i)14-s + (0.0484 + 0.132i)15-s + (2.61 + 0.953i)16-s + (1.13 − 3.12i)17-s + ⋯ |
| L(s) = 1 | + (−0.558 − 0.666i)2-s + (0.0164 + 0.0931i)3-s + (0.0423 − 0.240i)4-s + (0.380 − 0.0670i)5-s + (0.0528 − 0.0630i)6-s + (−0.374 − 0.927i)7-s + (−0.936 + 0.540i)8-s + (0.931 − 0.338i)9-s + (−0.257 − 0.215i)10-s + 0.577·11-s + 0.0230·12-s + (−1.10 − 0.929i)13-s + (−0.408 + 0.767i)14-s + (0.0124 + 0.0343i)15-s + (0.654 + 0.238i)16-s + (0.275 − 0.758i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.130 + 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.130 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.567807 - 0.647376i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.567807 - 0.647376i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (0.990 + 2.45i)T \) |
| 19 | \( 1 + (1.35 - 4.14i)T \) |
| good | 2 | \( 1 + (0.790 + 0.942i)T + (-0.347 + 1.96i)T^{2} \) |
| 3 | \( 1 + (-0.0284 - 0.161i)T + (-2.81 + 1.02i)T^{2} \) |
| 5 | \( 1 + (-0.850 + 0.150i)T + (4.69 - 1.71i)T^{2} \) |
| 11 | \( 1 - 1.91T + 11T^{2} \) |
| 13 | \( 1 + (3.99 + 3.35i)T + (2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-1.13 + 3.12i)T + (-13.0 - 10.9i)T^{2} \) |
| 23 | \( 1 + (-6.40 - 5.37i)T + (3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (-5.72 - 1.00i)T + (27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (-3.98 - 6.90i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-6.34 + 3.66i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (2.91 - 2.44i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (7.28 + 2.64i)T + (32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (-2.57 - 7.08i)T + (-36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 + (0.0763 + 0.0134i)T + (49.8 + 18.1i)T^{2} \) |
| 59 | \( 1 + (14.3 + 5.22i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-5.19 + 6.18i)T + (-10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (-0.248 + 0.296i)T + (-11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (1.73 - 4.76i)T + (-54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (0.264 - 0.0465i)T + (68.5 - 24.9i)T^{2} \) |
| 79 | \( 1 + (-4.31 + 11.8i)T + (-60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (-1.26 - 0.730i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (0.101 - 0.577i)T + (-83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (0.233 + 1.32i)T + (-91.1 + 33.1i)T^{2} \) |
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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
−12.79965390458888680264564652362, −11.89292036231381542824137582787, −10.60401898797638415396505376799, −9.907762610539450046762993133347, −9.350398059067801320813917409519, −7.65880664521986832547100427020, −6.49593806353120575008512833835, −4.98848166177141401832552077236, −3.22867693129922796033313959570, −1.22872827346547182144329413059,
2.51923380084223687787619336431, 4.52044482192329505200829147488, 6.30584607830949096830702616677, 6.96938022929424039428778143396, 8.280058975165811635823158498516, 9.262895978559690043760307402320, 10.02710741227788421745164428713, 11.73129680653271605530664328806, 12.49490075238974148368238188671, 13.44031543136381963161714842402
