| L(s) = 1 | + (0.491 + 1.32i)2-s + (−0.463 − 0.324i)3-s + (−1.51 + 1.30i)4-s + (2.08 + 0.182i)5-s + (0.202 − 0.773i)6-s + (2.46 + 1.42i)7-s + (−2.47 − 1.37i)8-s + (−0.916 − 2.51i)9-s + (0.781 + 2.85i)10-s + (−0.910 + 3.39i)11-s + (1.12 − 0.111i)12-s + (3.98 + 5.68i)13-s + (−0.675 + 3.96i)14-s + (−0.906 − 0.760i)15-s + (0.606 − 3.95i)16-s + (−0.157 − 0.0575i)17-s + ⋯ |
| L(s) = 1 | + (0.347 + 0.937i)2-s + (−0.267 − 0.187i)3-s + (−0.758 + 0.651i)4-s + (0.931 + 0.0814i)5-s + (0.0827 − 0.315i)6-s + (0.930 + 0.537i)7-s + (−0.874 − 0.485i)8-s + (−0.305 − 0.839i)9-s + (0.247 + 0.901i)10-s + (−0.274 + 1.02i)11-s + (0.325 − 0.0320i)12-s + (1.10 + 1.57i)13-s + (−0.180 + 1.05i)14-s + (−0.233 − 0.196i)15-s + (0.151 − 0.988i)16-s + (−0.0383 − 0.0139i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0718 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0718 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.07077 + 1.15065i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.07077 + 1.15065i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.491 - 1.32i)T \) |
| 19 | \( 1 + (0.574 + 4.32i)T \) |
| good | 3 | \( 1 + (0.463 + 0.324i)T + (1.02 + 2.81i)T^{2} \) |
| 5 | \( 1 + (-2.08 - 0.182i)T + (4.92 + 0.868i)T^{2} \) |
| 7 | \( 1 + (-2.46 - 1.42i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (0.910 - 3.39i)T + (-9.52 - 5.5i)T^{2} \) |
| 13 | \( 1 + (-3.98 - 5.68i)T + (-4.44 + 12.2i)T^{2} \) |
| 17 | \( 1 + (0.157 + 0.0575i)T + (13.0 + 10.9i)T^{2} \) |
| 23 | \( 1 + (3.82 - 4.56i)T + (-3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (-4.08 + 8.75i)T + (-18.6 - 22.2i)T^{2} \) |
| 31 | \( 1 + (1.60 - 2.78i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (0.980 + 0.980i)T + 37iT^{2} \) |
| 41 | \( 1 + (-4.30 + 0.758i)T + (38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (-4.19 - 0.367i)T + (42.3 + 7.46i)T^{2} \) |
| 47 | \( 1 + (0.436 - 0.158i)T + (36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 + (1.11 + 12.7i)T + (-52.1 + 9.20i)T^{2} \) |
| 59 | \( 1 + (5.24 + 11.2i)T + (-37.9 + 45.1i)T^{2} \) |
| 61 | \( 1 + (-14.0 + 1.23i)T + (60.0 - 10.5i)T^{2} \) |
| 67 | \( 1 + (-1.54 + 3.30i)T + (-43.0 - 51.3i)T^{2} \) |
| 71 | \( 1 + (4.26 + 5.08i)T + (-12.3 + 69.9i)T^{2} \) |
| 73 | \( 1 + (1.12 - 0.198i)T + (68.5 - 24.9i)T^{2} \) |
| 79 | \( 1 + (2.45 + 13.9i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (-3.08 - 11.5i)T + (-71.8 + 41.5i)T^{2} \) |
| 89 | \( 1 + (7.50 + 1.32i)T + (83.6 + 30.4i)T^{2} \) |
| 97 | \( 1 + (2.26 + 0.825i)T + (74.3 + 62.3i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.96161128999402810943543373706, −11.38459556185692092604970337365, −9.686303443001299671505960867938, −9.101486486438100708781571493304, −8.078337997335315296849752690326, −6.79252821465525317808706749498, −6.14945767183293752404999707880, −5.15615251482515594640236596320, −4.03398071137800966807147391283, −2.03815988659492559649363152993,
1.27747320504592204322938967856, 2.79194611399992352675688127496, 4.21670808297023143575789621573, 5.55925364502758058087335101237, 5.82857540153982445672076047920, 8.042470415874600563267908438699, 8.616500828513219975918684184594, 10.14586147217927103858149974099, 10.68243775496625577255799808081, 11.11169352090502172489052589148
