L(s) = 1 | + (−0.309 − 0.951i)2-s + (−0.649 + 0.471i)3-s + (−0.809 + 0.587i)4-s + (0.192 − 2.22i)5-s + (0.649 + 0.471i)6-s + 7-s + (0.809 + 0.587i)8-s + (−0.727 + 2.23i)9-s + (−2.17 + 0.505i)10-s + (−1.92 − 5.91i)11-s + (0.248 − 0.763i)12-s + (0.707 − 2.17i)13-s + (−0.309 − 0.951i)14-s + (0.926 + 1.53i)15-s + (0.309 − 0.951i)16-s + (−2.48 − 1.80i)17-s + ⋯ |
L(s) = 1 | + (−0.218 − 0.672i)2-s + (−0.375 + 0.272i)3-s + (−0.404 + 0.293i)4-s + (0.0861 − 0.996i)5-s + (0.265 + 0.192i)6-s + 0.377·7-s + (0.286 + 0.207i)8-s + (−0.242 + 0.746i)9-s + (−0.688 + 0.159i)10-s + (−0.579 − 1.78i)11-s + (0.0716 − 0.220i)12-s + (0.196 − 0.603i)13-s + (−0.0825 − 0.254i)14-s + (0.239 + 0.397i)15-s + (0.0772 − 0.237i)16-s + (−0.601 − 0.437i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.672 + 0.739i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.672 + 0.739i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.335047 - 0.757519i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.335047 - 0.757519i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.192 + 2.22i)T \) |
| 7 | \( 1 - T \) |
good | 3 | \( 1 + (0.649 - 0.471i)T + (0.927 - 2.85i)T^{2} \) |
| 11 | \( 1 + (1.92 + 5.91i)T + (-8.89 + 6.46i)T^{2} \) |
| 13 | \( 1 + (-0.707 + 2.17i)T + (-10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (2.48 + 1.80i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (-3.39 - 2.46i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (2.47 + 7.60i)T + (-18.6 + 13.5i)T^{2} \) |
| 29 | \( 1 + (2.89 - 2.10i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (3.45 + 2.50i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (-1.89 + 5.84i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (2.68 - 8.25i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 - 10.2T + 43T^{2} \) |
| 47 | \( 1 + (-5.31 + 3.85i)T + (14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (0.0502 - 0.0365i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (0.791 - 2.43i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (-3.29 - 10.1i)T + (-49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + (-3.63 - 2.64i)T + (20.7 + 63.7i)T^{2} \) |
| 71 | \( 1 + (-12.1 + 8.84i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-2.32 - 7.16i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (1.05 - 0.769i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (-6.05 - 4.39i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + (-3.39 - 10.4i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (-6.93 + 5.04i)T + (29.9 - 92.2i)T^{2} \) |
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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
−11.01684580176654770318297224181, −10.54335005214008373894701731639, −9.293789017326786195536709625170, −8.393728562854779623018388022407, −7.86139989027242440871696440116, −5.81752143650046880813313328698, −5.22234102650256902413351716249, −4.02199987572604899725748799934, −2.50018812479466657026811931826, −0.64622272457743794333753295966,
1.99343071283918862483228769429, 3.81419686056319707289065818562, 5.16678900769758035724171024963, 6.23619030536104351446610895326, 7.14999073452508278405624356927, 7.62862477861308745637805211017, 9.178048070268195054951086065768, 9.819329347608943887433867306327, 10.92922637368059727266578152101, 11.72138130314708918384119640171