L(s) = 1 | + (−0.5 + 0.866i)2-s + (−0.866 + 0.5i)3-s + (−0.499 − 0.866i)4-s + (−1.81 + 3.14i)5-s − 0.999i·6-s + (−1.58 + 2.11i)7-s + 0.999·8-s + (0.499 − 0.866i)9-s + (−1.81 − 3.14i)10-s + (0.401 − 0.231i)11-s + (0.866 + 0.499i)12-s + 6.88i·13-s + (−1.04 − 2.43i)14-s − 3.63i·15-s + (−0.5 + 0.866i)16-s + (1.54 + 2.68i)17-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.499 + 0.288i)3-s + (−0.249 − 0.433i)4-s + (−0.812 + 1.40i)5-s − 0.408i·6-s + (−0.598 + 0.800i)7-s + 0.353·8-s + (0.166 − 0.288i)9-s + (−0.574 − 0.995i)10-s + (0.121 − 0.0699i)11-s + (0.249 + 0.144i)12-s + 1.90i·13-s + (−0.278 − 0.649i)14-s − 0.938i·15-s + (−0.125 + 0.216i)16-s + (0.375 + 0.649i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.287 + 0.957i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.287 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.295253 - 0.396879i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.295253 - 0.396879i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 3 | \( 1 + (0.866 - 0.5i)T \) |
| 7 | \( 1 + (1.58 - 2.11i)T \) |
| 23 | \( 1 + (2.20 + 4.25i)T \) |
good | 5 | \( 1 + (1.81 - 3.14i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-0.401 + 0.231i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 6.88iT - 13T^{2} \) |
| 17 | \( 1 + (-1.54 - 2.68i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (2.90 - 5.02i)T + (-9.5 - 16.4i)T^{2} \) |
| 29 | \( 1 - 3.72T + 29T^{2} \) |
| 31 | \( 1 + (2.45 - 1.41i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-6.15 - 3.55i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 8.56iT - 41T^{2} \) |
| 43 | \( 1 - 2.69iT - 43T^{2} \) |
| 47 | \( 1 + (2.16 + 1.24i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-4.36 + 2.51i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-9.36 + 5.40i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.47 + 2.55i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (4.25 - 2.45i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 8.05T + 71T^{2} \) |
| 73 | \( 1 + (7.09 - 4.09i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (9.37 + 5.41i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 0.215T + 83T^{2} \) |
| 89 | \( 1 + (5.36 - 9.29i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 12.0T + 97T^{2} \) |
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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
−10.42754427913425112864405660887, −9.903815066829298264316685106534, −8.857241287444508295717099970559, −8.101707096813707300719576653644, −6.92304359554682527610012210827, −6.52085703587985626266259713984, −5.82438075741588575599232237955, −4.34982965389906593499238209793, −3.60599498892022211191440330358, −2.16806060606369803741975755813,
0.34657035581856785770032735838, 1.02399818425044024716041223504, 2.92972358727500172566516224760, 4.04683327068329929331788550097, 4.86888432271740035019962183082, 5.81540594906156048340260675196, 7.21963491629532640741615254169, 7.82104189093581464093053656794, 8.581788106649190454108207793092, 9.547753262312270662040687858983