| L(s) = 1 | + (1.27 + 0.609i)2-s + (0.884 + 0.187i)3-s + (1.25 + 1.55i)4-s + (−1.21 − 2.09i)5-s + (1.01 + 0.779i)6-s + (−0.336 − 0.754i)7-s + (0.652 + 2.75i)8-s + (−1.99 − 0.887i)9-s + (−0.265 − 3.41i)10-s + (−0.549 + 5.22i)11-s + (0.817 + 1.61i)12-s + (−3.26 − 2.93i)13-s + (0.0316 − 1.16i)14-s + (−0.676 − 2.08i)15-s + (−0.845 + 3.90i)16-s + (1.25 − 0.132i)17-s + ⋯ |
| L(s) = 1 | + (0.902 + 0.431i)2-s + (0.510 + 0.108i)3-s + (0.627 + 0.778i)4-s + (−0.541 − 0.938i)5-s + (0.413 + 0.318i)6-s + (−0.127 − 0.285i)7-s + (0.230 + 0.972i)8-s + (−0.664 − 0.295i)9-s + (−0.0840 − 1.08i)10-s + (−0.165 + 1.57i)11-s + (0.236 + 0.465i)12-s + (−0.904 − 0.814i)13-s + (0.00845 − 0.312i)14-s + (−0.174 − 0.537i)15-s + (−0.211 + 0.977i)16-s + (0.305 − 0.0320i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.883 - 0.468i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.883 - 0.468i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.66587 + 0.414428i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.66587 + 0.414428i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.27 - 0.609i)T \) |
| 31 | \( 1 + (5.56 + 0.207i)T \) |
| good | 3 | \( 1 + (-0.884 - 0.187i)T + (2.74 + 1.22i)T^{2} \) |
| 5 | \( 1 + (1.21 + 2.09i)T + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (0.336 + 0.754i)T + (-4.68 + 5.20i)T^{2} \) |
| 11 | \( 1 + (0.549 - 5.22i)T + (-10.7 - 2.28i)T^{2} \) |
| 13 | \( 1 + (3.26 + 2.93i)T + (1.35 + 12.9i)T^{2} \) |
| 17 | \( 1 + (-1.25 + 0.132i)T + (16.6 - 3.53i)T^{2} \) |
| 19 | \( 1 + (-0.980 + 0.882i)T + (1.98 - 18.8i)T^{2} \) |
| 23 | \( 1 + (-3.19 + 2.31i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (-9.46 - 3.07i)T + (23.4 + 17.0i)T^{2} \) |
| 37 | \( 1 + (0.546 + 0.315i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (5.23 - 1.11i)T + (37.4 - 16.6i)T^{2} \) |
| 43 | \( 1 + (-2.09 - 2.32i)T + (-4.49 + 42.7i)T^{2} \) |
| 47 | \( 1 + (-5.84 + 1.89i)T + (38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (1.71 - 3.85i)T + (-35.4 - 39.3i)T^{2} \) |
| 59 | \( 1 + (2.62 - 12.3i)T + (-53.8 - 23.9i)T^{2} \) |
| 61 | \( 1 + 2.25iT - 61T^{2} \) |
| 67 | \( 1 + (8.98 - 5.19i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (0.572 - 1.28i)T + (-47.5 - 52.7i)T^{2} \) |
| 73 | \( 1 + (-8.31 - 0.873i)T + (71.4 + 15.1i)T^{2} \) |
| 79 | \( 1 + (0.824 + 7.84i)T + (-77.2 + 16.4i)T^{2} \) |
| 83 | \( 1 + (-11.4 + 2.43i)T + (75.8 - 33.7i)T^{2} \) |
| 89 | \( 1 + (-3.29 + 4.53i)T + (-27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (1.54 + 1.12i)T + (29.9 + 92.2i)T^{2} \) |
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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
−13.48900146734234238681783430450, −12.37755425322592794259861277286, −12.12353601715623089312819851961, −10.41840705577690628815542645547, −9.004596970684804539961018511750, −7.947871160193508946873065845232, −7.01992433382144532864441454127, −5.26897798859359639253594923291, −4.36968761683461012613274977816, −2.83086047425379760014462091350,
2.65971390891847386325635256010, 3.51795603551548377456530275963, 5.32251388992984100977046796688, 6.57229492995677476349235705075, 7.78979313018909250266429406235, 9.181732731713441310029097694430, 10.64204448946411005887571691904, 11.36772648353877807717390630673, 12.20872551921970940531879733353, 13.66906716407651037496321317625
