| L(s) = 1 | + (1.08 + 1.20i)2-s + (0.913 + 0.406i)3-s + (−0.0646 + 0.614i)4-s + (−2.18 − 0.464i)5-s + (0.5 + 1.53i)6-s + (−2.53 − 0.768i)7-s + (1.80 − 1.31i)8-s + (−1.33 − 1.48i)9-s + (−1.80 − 3.13i)10-s + (2.24 + 2.43i)11-s + (−0.309 + 0.535i)12-s + (−1.80 + 5.56i)13-s + (−1.81 − 3.87i)14-s + (−1.80 − 1.31i)15-s + (4.74 + 1.00i)16-s + (2.16 − 2.40i)17-s + ⋯ |
| L(s) = 1 | + (0.765 + 0.850i)2-s + (0.527 + 0.234i)3-s + (−0.0323 + 0.307i)4-s + (−0.978 − 0.207i)5-s + (0.204 + 0.628i)6-s + (−0.956 − 0.290i)7-s + (0.639 − 0.464i)8-s + (−0.446 − 0.495i)9-s + (−0.572 − 0.990i)10-s + (0.677 + 0.735i)11-s + (−0.0892 + 0.154i)12-s + (−0.501 + 1.54i)13-s + (−0.485 − 1.03i)14-s + (−0.467 − 0.339i)15-s + (1.18 + 0.252i)16-s + (0.525 − 0.583i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 - 0.766i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.17877 + 0.549838i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.17877 + 0.549838i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (2.53 + 0.768i)T \) |
| 11 | \( 1 + (-2.24 - 2.43i)T \) |
| good | 2 | \( 1 + (-1.08 - 1.20i)T + (-0.209 + 1.98i)T^{2} \) |
| 3 | \( 1 + (-0.913 - 0.406i)T + (2.00 + 2.22i)T^{2} \) |
| 5 | \( 1 + (2.18 + 0.464i)T + (4.56 + 2.03i)T^{2} \) |
| 13 | \( 1 + (1.80 - 5.56i)T + (-10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (-2.16 + 2.40i)T + (-1.77 - 16.9i)T^{2} \) |
| 19 | \( 1 + (0.119 + 1.13i)T + (-18.5 + 3.95i)T^{2} \) |
| 23 | \( 1 + (-0.881 + 1.52i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-6.35 - 4.61i)T + (8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (4.28 - 0.911i)T + (28.3 - 12.6i)T^{2} \) |
| 37 | \( 1 + (5.56 - 2.47i)T + (24.7 - 27.4i)T^{2} \) |
| 41 | \( 1 + (-1.92 + 1.40i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 - 0.145T + 43T^{2} \) |
| 47 | \( 1 + (0.0493 + 0.469i)T + (-45.9 + 9.77i)T^{2} \) |
| 53 | \( 1 + (5.95 - 1.26i)T + (48.4 - 21.5i)T^{2} \) |
| 59 | \( 1 + (-1.01 + 9.65i)T + (-57.7 - 12.2i)T^{2} \) |
| 61 | \( 1 + (-10.7 - 2.27i)T + (55.7 + 24.8i)T^{2} \) |
| 67 | \( 1 + (4.19 + 7.25i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-0.236 - 0.726i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (1.13 - 10.7i)T + (-71.4 - 15.1i)T^{2} \) |
| 79 | \( 1 + (8.50 + 9.44i)T + (-8.25 + 78.5i)T^{2} \) |
| 83 | \( 1 + (2.78 + 8.55i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (0.763 - 1.32i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (1.19 - 3.66i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−14.49989623384955257138142009198, −14.08820812244744539791107966052, −12.60439497603582223571133210050, −11.74312758081648122614041317013, −9.931481762188244287098479287834, −8.926601843641222965865453216308, −7.27788338345490568685465870503, −6.53527789999421348087676857840, −4.64750587798775663100186465814, −3.61803503097221804920034954135,
2.88548250612170230312294576514, 3.70075312541349697491267104976, 5.63087306893822093115757178744, 7.57662410744124785376689809950, 8.464293281589435752703645934815, 10.22729004951418089364125017119, 11.33750756686325643279403772417, 12.29608906427511164465254855062, 13.07610442411766967616775429504, 14.10106222043813543171211117823
