| L(s) = 1 | + (−0.831 + 2.55i)2-s + (1.38 + 0.295i)3-s + (−4.23 − 3.07i)4-s + (−0.304 − 0.526i)5-s + (−1.90 + 3.30i)6-s + (−1.57 + 0.702i)7-s + (7.04 − 5.11i)8-s + (−0.900 − 0.401i)9-s + (1.60 − 0.340i)10-s + (−0.139 + 1.33i)11-s + (−4.97 − 5.52i)12-s + (−2.45 + 2.72i)13-s + (−0.485 − 4.62i)14-s + (−0.266 − 0.821i)15-s + (4.00 + 12.3i)16-s + (−0.288 − 2.74i)17-s + ⋯ |
| L(s) = 1 | + (−0.587 + 1.80i)2-s + (0.801 + 0.170i)3-s + (−2.11 − 1.53i)4-s + (−0.136 − 0.235i)5-s + (−0.779 + 1.34i)6-s + (−0.596 + 0.265i)7-s + (2.49 − 1.80i)8-s + (−0.300 − 0.133i)9-s + (0.506 − 0.107i)10-s + (−0.0421 + 0.401i)11-s + (−1.43 − 1.59i)12-s + (−0.681 + 0.756i)13-s + (−0.129 − 1.23i)14-s + (−0.0688 − 0.212i)15-s + (1.00 + 3.07i)16-s + (−0.0700 − 0.666i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 + 0.191i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.981 + 0.191i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.455600 - 0.0440524i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.455600 - 0.0440524i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 31 | \( 1 \) |
| good | 2 | \( 1 + (0.831 - 2.55i)T + (-1.61 - 1.17i)T^{2} \) |
| 3 | \( 1 + (-1.38 - 0.295i)T + (2.74 + 1.22i)T^{2} \) |
| 5 | \( 1 + (0.304 + 0.526i)T + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (1.57 - 0.702i)T + (4.68 - 5.20i)T^{2} \) |
| 11 | \( 1 + (0.139 - 1.33i)T + (-10.7 - 2.28i)T^{2} \) |
| 13 | \( 1 + (2.45 - 2.72i)T + (-1.35 - 12.9i)T^{2} \) |
| 17 | \( 1 + (0.288 + 2.74i)T + (-16.6 + 3.53i)T^{2} \) |
| 19 | \( 1 + (1.71 + 1.90i)T + (-1.98 + 18.8i)T^{2} \) |
| 23 | \( 1 + (0.436 - 0.316i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (-2.51 + 7.73i)T + (-23.4 - 17.0i)T^{2} \) |
| 37 | \( 1 + (-3.87 + 6.70i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.101 + 0.0216i)T + (37.4 - 16.6i)T^{2} \) |
| 43 | \( 1 + (-2.01 - 2.23i)T + (-4.49 + 42.7i)T^{2} \) |
| 47 | \( 1 + (2.07 + 6.39i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (2.55 + 1.13i)T + (35.4 + 39.3i)T^{2} \) |
| 59 | \( 1 + (0.456 + 0.0969i)T + (53.8 + 23.9i)T^{2} \) |
| 61 | \( 1 - 5.11T + 61T^{2} \) |
| 67 | \( 1 + (4.14 + 7.18i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-4.34 - 1.93i)T + (47.5 + 52.7i)T^{2} \) |
| 73 | \( 1 + (-0.784 + 7.46i)T + (-71.4 - 15.1i)T^{2} \) |
| 79 | \( 1 + (1.01 + 9.64i)T + (-77.2 + 16.4i)T^{2} \) |
| 83 | \( 1 + (16.3 - 3.46i)T + (75.8 - 33.7i)T^{2} \) |
| 89 | \( 1 + (12.3 + 9.00i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (-1.03 - 0.751i)T + (29.9 + 92.2i)T^{2} \) |
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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
−9.452511535126495688852571780887, −9.091065263168730784282293360608, −8.269265103248353905110477985169, −7.56953485206478298757245421037, −6.69562273252522620185342251539, −6.01659759664385156312117696340, −4.88422923388782944255333867421, −4.12023819927824853898544848854, −2.49617633265648994154531445029, −0.24044853015217187104451254767,
1.43403486384627568646260540382, 2.78375159415320519173913001216, 3.14316607273529950231530229240, 4.13867472989305890066502206861, 5.42981861620699183815893398249, 7.03574009607051237368077650647, 8.143324940955890113343970579498, 8.472540239467786468605754308149, 9.444158483242262838181818373067, 10.08291531473796552745715000424
