| L(s) = 1 | + (−0.188 + 1.40i)2-s + (2.71 + 0.390i)3-s + (−1.92 − 0.527i)4-s + (0.232 + 0.791i)5-s + (−1.05 + 3.73i)6-s + (−2.90 − 3.35i)7-s + (1.10 − 2.60i)8-s + (4.34 + 1.27i)9-s + (−1.15 + 0.176i)10-s + (−1.51 − 0.972i)11-s + (−5.03 − 2.18i)12-s + (−3.65 + 4.22i)13-s + (5.24 − 3.43i)14-s + (0.321 + 2.23i)15-s + (3.44 + 2.03i)16-s + (1.07 − 0.491i)17-s + ⋯ |
| L(s) = 1 | + (−0.133 + 0.991i)2-s + (1.56 + 0.225i)3-s + (−0.964 − 0.263i)4-s + (0.103 + 0.353i)5-s + (−0.431 + 1.52i)6-s + (−1.09 − 1.26i)7-s + (0.389 − 0.920i)8-s + (1.44 + 0.424i)9-s + (−0.364 + 0.0559i)10-s + (−0.456 − 0.293i)11-s + (−1.45 − 0.630i)12-s + (−1.01 + 1.17i)13-s + (1.40 − 0.919i)14-s + (0.0831 + 0.578i)15-s + (0.860 + 0.508i)16-s + (0.260 − 0.119i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.999310 + 0.665006i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.999310 + 0.665006i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.188 - 1.40i)T \) |
| 23 | \( 1 + (-4.16 + 2.37i)T \) |
| good | 3 | \( 1 + (-2.71 - 0.390i)T + (2.87 + 0.845i)T^{2} \) |
| 5 | \( 1 + (-0.232 - 0.791i)T + (-4.20 + 2.70i)T^{2} \) |
| 7 | \( 1 + (2.90 + 3.35i)T + (-0.996 + 6.92i)T^{2} \) |
| 11 | \( 1 + (1.51 + 0.972i)T + (4.56 + 10.0i)T^{2} \) |
| 13 | \( 1 + (3.65 - 4.22i)T + (-1.85 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-1.07 + 0.491i)T + (11.1 - 12.8i)T^{2} \) |
| 19 | \( 1 + (0.454 - 0.994i)T + (-12.4 - 14.3i)T^{2} \) |
| 29 | \( 1 + (-1.35 - 2.95i)T + (-18.9 + 21.9i)T^{2} \) |
| 31 | \( 1 + (1.36 - 0.196i)T + (29.7 - 8.73i)T^{2} \) |
| 37 | \( 1 + (0.0441 - 0.150i)T + (-31.1 - 20.0i)T^{2} \) |
| 41 | \( 1 + (3.74 - 1.09i)T + (34.4 - 22.1i)T^{2} \) |
| 43 | \( 1 + (1.05 - 7.34i)T + (-41.2 - 12.1i)T^{2} \) |
| 47 | \( 1 + 1.42iT - 47T^{2} \) |
| 53 | \( 1 + (-8.39 + 7.27i)T + (7.54 - 52.4i)T^{2} \) |
| 59 | \( 1 + (-2.52 - 2.18i)T + (8.39 + 58.3i)T^{2} \) |
| 61 | \( 1 + (8.75 - 1.25i)T + (58.5 - 17.1i)T^{2} \) |
| 67 | \( 1 + (7.17 - 4.60i)T + (27.8 - 60.9i)T^{2} \) |
| 71 | \( 1 + (-0.0394 - 0.0613i)T + (-29.4 + 64.5i)T^{2} \) |
| 73 | \( 1 + (-3.13 + 6.86i)T + (-47.8 - 55.1i)T^{2} \) |
| 79 | \( 1 + (3.24 - 3.74i)T + (-11.2 - 78.1i)T^{2} \) |
| 83 | \( 1 + (-6.48 - 1.90i)T + (69.8 + 44.8i)T^{2} \) |
| 89 | \( 1 + (2.37 + 0.341i)T + (85.3 + 25.0i)T^{2} \) |
| 97 | \( 1 + (4.11 + 14.0i)T + (-81.6 + 52.4i)T^{2} \) |
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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
−14.32839605370923586302059650262, −13.66730908898401044745434037466, −12.81772589127261863314432951114, −10.37318791918613251792657634695, −9.665151616897344428205987501625, −8.686833285951498551973929048092, −7.41048123416196928902652815313, −6.72763138875336100883190562135, −4.47706068132413489801929012009, −3.17547779243313116465568045150,
2.44900881774292823826180844307, 3.23844201427824931520499531532, 5.27042056330420996546759884263, 7.55291841430764043580511042279, 8.725261678775874929678226343139, 9.352332577544927892275882439491, 10.26732171041274888185349228590, 12.19016416381142729849396976943, 12.83317183559873197849357201378, 13.47792792029533719990933725482
