| L(s) = 1 | + (0.156 + 0.987i)2-s + (−1.34 + 1.09i)3-s + (−0.951 + 0.309i)4-s + (−1.59 − 1.56i)5-s + (−1.28 − 1.15i)6-s + (−3.45 + 3.45i)7-s + (−0.453 − 0.891i)8-s + (0.615 − 2.93i)9-s + (1.29 − 1.82i)10-s + (−1.48 + 2.05i)11-s + (0.941 − 1.45i)12-s + (3.77 + 0.597i)13-s + (−3.95 − 2.86i)14-s + (3.85 + 0.354i)15-s + (0.809 − 0.587i)16-s + (−4.27 + 2.18i)17-s + ⋯ |
| L(s) = 1 | + (0.110 + 0.698i)2-s + (−0.776 + 0.630i)3-s + (−0.475 + 0.154i)4-s + (−0.715 − 0.698i)5-s + (−0.526 − 0.472i)6-s + (−1.30 + 1.30i)7-s + (−0.160 − 0.315i)8-s + (0.205 − 0.978i)9-s + (0.408 − 0.576i)10-s + (−0.449 + 0.618i)11-s + (0.271 − 0.419i)12-s + (1.04 + 0.165i)13-s + (−1.05 − 0.767i)14-s + (0.995 + 0.0915i)15-s + (0.202 − 0.146i)16-s + (−1.03 + 0.528i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 + 0.108i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.994 + 0.108i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0233207 - 0.429369i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0233207 - 0.429369i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.156 - 0.987i)T \) |
| 3 | \( 1 + (1.34 - 1.09i)T \) |
| 5 | \( 1 + (1.59 + 1.56i)T \) |
| good | 7 | \( 1 + (3.45 - 3.45i)T - 7iT^{2} \) |
| 11 | \( 1 + (1.48 - 2.05i)T + (-3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (-3.77 - 0.597i)T + (12.3 + 4.01i)T^{2} \) |
| 17 | \( 1 + (4.27 - 2.18i)T + (9.99 - 13.7i)T^{2} \) |
| 19 | \( 1 + (-0.596 - 0.193i)T + (15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (-0.643 + 0.101i)T + (21.8 - 7.10i)T^{2} \) |
| 29 | \( 1 + (-2.50 - 7.72i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-0.501 + 1.54i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (0.576 - 3.64i)T + (-35.1 - 11.4i)T^{2} \) |
| 41 | \( 1 + (3.69 + 5.08i)T + (-12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + (0.951 + 0.951i)T + 43iT^{2} \) |
| 47 | \( 1 + (-1.95 + 3.82i)T + (-27.6 - 38.0i)T^{2} \) |
| 53 | \( 1 + (-0.456 - 0.232i)T + (31.1 + 42.8i)T^{2} \) |
| 59 | \( 1 + (8.96 - 6.51i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (-3.22 - 2.34i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + (-1.95 - 3.82i)T + (-39.3 + 54.2i)T^{2} \) |
| 71 | \( 1 + (-0.917 + 0.297i)T + (57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (1.21 + 7.65i)T + (-69.4 + 22.5i)T^{2} \) |
| 79 | \( 1 + (-0.139 + 0.0451i)T + (63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (3.59 + 7.05i)T + (-48.7 + 67.1i)T^{2} \) |
| 89 | \( 1 + (-8.93 - 6.49i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (13.5 + 6.89i)T + (57.0 + 78.4i)T^{2} \) |
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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
−13.23688121024158547954977265038, −12.54463077262420068158164052607, −11.77774637240762617482440896071, −10.43113934111836814162108352460, −9.150644778766688800839965646816, −8.645349647167117805301658721887, −6.86134044538330187131255294823, −5.91172497054464206357880643629, −4.86938384071581806598360946666, −3.56449208069968336424818034799,
0.44232339022431889338660688033, 3.04947510845837530172068145013, 4.27824691742695423406646585223, 6.15603270248256915624063541825, 6.97004334842027033779132187350, 8.142430408441476470419902499457, 9.872597134352462531556777346911, 10.87011697708816053308855499831, 11.23401203731258243871831551906, 12.50096356556180644439117485279
