| 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
−11.72138130314708918384119640171, −10.92922637368059727266578152101, −9.819329347608943887433867306327, −9.178048070268195054951086065768, −7.62862477861308745637805211017, −7.14999073452508278405624356927, −6.23619030536104351446610895326, −5.16678900769758035724171024963, −3.81419686056319707289065818562, −1.99343071283918862483228769429,
0.64622272457743794333753295966, 2.50018812479466657026811931826, 4.02199987572604899725748799934, 5.22234102650256902413351716249, 5.81752143650046880813313328698, 7.86139989027242440871696440116, 8.393728562854779623018388022407, 9.293789017326786195536709625170, 10.54335005214008373894701731639, 11.01684580176654770318297224181
