L(s) = 1 | + (0.309 + 0.951i)2-s + (0.521 − 0.378i)3-s + (−0.809 + 0.587i)4-s + (−0.491 − 2.18i)5-s + (0.521 + 0.378i)6-s − 2.52·7-s + (−0.809 − 0.587i)8-s + (−0.798 + 2.45i)9-s + (1.92 − 1.14i)10-s + (0.688 + 2.11i)11-s + (−0.199 + 0.613i)12-s + (−0.343 + 1.05i)13-s + (−0.781 − 2.40i)14-s + (−1.08 − 0.951i)15-s + (0.309 − 0.951i)16-s + (5.55 + 4.03i)17-s + ⋯ |
L(s) = 1 | + (0.218 + 0.672i)2-s + (0.301 − 0.218i)3-s + (−0.404 + 0.293i)4-s + (−0.219 − 0.975i)5-s + (0.212 + 0.154i)6-s − 0.956·7-s + (−0.286 − 0.207i)8-s + (−0.266 + 0.819i)9-s + (0.608 − 0.360i)10-s + (0.207 + 0.638i)11-s + (−0.0575 + 0.176i)12-s + (−0.0952 + 0.293i)13-s + (−0.208 − 0.643i)14-s + (−0.279 − 0.245i)15-s + (0.0772 − 0.237i)16-s + (1.34 + 0.978i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.484 - 0.874i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.484 - 0.874i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.612612 + 1.03945i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.612612 + 1.03945i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.309 - 0.951i)T \) |
| 5 | \( 1 + (0.491 + 2.18i)T \) |
| 19 | \( 1 + (-0.809 - 0.587i)T \) |
good | 3 | \( 1 + (-0.521 + 0.378i)T + (0.927 - 2.85i)T^{2} \) |
| 7 | \( 1 + 2.52T + 7T^{2} \) |
| 11 | \( 1 + (-0.688 - 2.11i)T + (-8.89 + 6.46i)T^{2} \) |
| 13 | \( 1 + (0.343 - 1.05i)T + (-10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (-5.55 - 4.03i)T + (5.25 + 16.1i)T^{2} \) |
| 23 | \( 1 + (-0.471 - 1.45i)T + (-18.6 + 13.5i)T^{2} \) |
| 29 | \( 1 + (5.23 - 3.80i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (0.0336 + 0.0244i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (1.11 - 3.41i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (1.40 - 4.31i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 - 7.28T + 43T^{2} \) |
| 47 | \( 1 + (1.11 - 0.813i)T + (14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (-0.355 + 0.258i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (3.99 - 12.2i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (2.54 + 7.82i)T + (-49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + (-3.05 - 2.22i)T + (20.7 + 63.7i)T^{2} \) |
| 71 | \( 1 + (-13.4 + 9.76i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-4.52 - 13.9i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (1.48 - 1.07i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (8.43 + 6.12i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + (3.76 + 11.5i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (12.4 - 9.01i)T + (29.9 - 92.2i)T^{2} \) |
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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
−9.965288447833867109967669208324, −9.393432011111530470296860559048, −8.474589799734658614238521134056, −7.79720770617070460942282786640, −7.06277224330677324360566522382, −5.91740000053705661320292571279, −5.21678696148416284020466125732, −4.17220022856253712162730102853, −3.19094706058657643659399205420, −1.56595437652287481005046068345,
0.51580345990424214524871048322, 2.60742841168933381191311159708, 3.32785195550131598575839866131, 3.86183069760243364161459475345, 5.47073264061361636835976229416, 6.24831290095765988227647778015, 7.16517818897223745272852269898, 8.169675992666122267743203574223, 9.426243034107965847884954989197, 9.627011704925234206177079431416