| L(s) = 1 | + (1.44 + 1.66i)2-s + (−0.841 − 0.540i)3-s + (−0.406 + 2.82i)4-s + (0.246 − 0.540i)5-s + (−0.313 − 2.18i)6-s + (−2.76 − 0.813i)7-s + (−1.58 + 1.01i)8-s + (0.415 + 0.909i)9-s + (1.25 − 0.368i)10-s + (2.87 − 3.32i)11-s + (1.87 − 2.15i)12-s + (−5.22 + 1.53i)13-s + (−2.64 − 5.78i)14-s + (−0.5 + 0.321i)15-s + (1.49 + 0.438i)16-s + (0.543 + 3.77i)17-s + ⋯ |
| L(s) = 1 | + (1.02 + 1.17i)2-s + (−0.485 − 0.312i)3-s + (−0.203 + 1.41i)4-s + (0.110 − 0.241i)5-s + (−0.128 − 0.890i)6-s + (−1.04 − 0.307i)7-s + (−0.560 + 0.360i)8-s + (0.138 + 0.303i)9-s + (0.397 − 0.116i)10-s + (0.868 − 1.00i)11-s + (0.539 − 0.623i)12-s + (−1.44 + 0.425i)13-s + (−0.706 − 1.54i)14-s + (−0.129 + 0.0829i)15-s + (0.373 + 0.109i)16-s + (0.131 + 0.916i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.450 - 0.892i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.450 - 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.03137 + 0.634963i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.03137 + 0.634963i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.841 + 0.540i)T \) |
| 23 | \( 1 + (2.38 + 4.16i)T \) |
| good | 2 | \( 1 + (-1.44 - 1.66i)T + (-0.284 + 1.97i)T^{2} \) |
| 5 | \( 1 + (-0.246 + 0.540i)T + (-3.27 - 3.77i)T^{2} \) |
| 7 | \( 1 + (2.76 + 0.813i)T + (5.88 + 3.78i)T^{2} \) |
| 11 | \( 1 + (-2.87 + 3.32i)T + (-1.56 - 10.8i)T^{2} \) |
| 13 | \( 1 + (5.22 - 1.53i)T + (10.9 - 7.02i)T^{2} \) |
| 17 | \( 1 + (-0.543 - 3.77i)T + (-16.3 + 4.78i)T^{2} \) |
| 19 | \( 1 + (-0.0164 + 0.114i)T + (-18.2 - 5.35i)T^{2} \) |
| 29 | \( 1 + (-0.782 - 5.44i)T + (-27.8 + 8.17i)T^{2} \) |
| 31 | \( 1 + (5.64 - 3.62i)T + (12.8 - 28.1i)T^{2} \) |
| 37 | \( 1 + (3.82 + 8.36i)T + (-24.2 + 27.9i)T^{2} \) |
| 41 | \( 1 + (-1.77 + 3.87i)T + (-26.8 - 30.9i)T^{2} \) |
| 43 | \( 1 + (-2.64 - 1.70i)T + (17.8 + 39.1i)T^{2} \) |
| 47 | \( 1 - 3.51T + 47T^{2} \) |
| 53 | \( 1 + (-9.41 - 2.76i)T + (44.5 + 28.6i)T^{2} \) |
| 59 | \( 1 + (-3.89 + 1.14i)T + (49.6 - 31.8i)T^{2} \) |
| 61 | \( 1 + (-9.95 + 6.39i)T + (25.3 - 55.4i)T^{2} \) |
| 67 | \( 1 + (4.70 + 5.43i)T + (-9.53 + 66.3i)T^{2} \) |
| 71 | \( 1 + (5.28 + 6.09i)T + (-10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (-0.543 + 3.77i)T + (-70.0 - 20.5i)T^{2} \) |
| 79 | \( 1 + (0.375 - 0.110i)T + (66.4 - 42.7i)T^{2} \) |
| 83 | \( 1 + (0.397 + 0.869i)T + (-54.3 + 62.7i)T^{2} \) |
| 89 | \( 1 + (5.05 + 3.24i)T + (36.9 + 80.9i)T^{2} \) |
| 97 | \( 1 + (6.52 - 14.2i)T + (-63.5 - 73.3i)T^{2} \) |
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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
−14.73464497853688095233579658106, −14.04140512984038268785685510517, −12.83463214980509905875041401969, −12.29301549413603898837190307228, −10.55250879471868652115352804462, −8.954122527018007768857573395621, −7.26098274593400918953174006432, −6.46079870070064049352366716025, −5.35306336464161833200999969309, −3.79690007635598834504112983659,
2.66114685994115016580552222729, 4.21836561575609531150688297118, 5.52877607906957651857956608863, 7.05874227787938681151726649831, 9.675039513374313742744963097593, 10.06367978190764606593214044983, 11.66875330960159442896435381657, 12.19630610226503590629105144958, 13.14887447930913709922329142011, 14.40610316436649677822953922558
