L(s) = 1 | + (0.309 + 0.951i)2-s + (1.89 − 1.37i)3-s + (−0.809 + 0.587i)4-s + (1.73 − 1.40i)5-s + (1.89 + 1.37i)6-s + 2.10·7-s + (−0.809 − 0.587i)8-s + (0.771 − 2.37i)9-s + (1.87 + 1.21i)10-s + (−0.871 − 2.68i)11-s + (−0.724 + 2.22i)12-s + (−1.13 + 3.49i)13-s + (0.650 + 2.00i)14-s + (1.34 − 5.06i)15-s + (0.309 − 0.951i)16-s + (2.68 + 1.95i)17-s + ⋯ |
L(s) = 1 | + (0.218 + 0.672i)2-s + (1.09 − 0.795i)3-s + (−0.404 + 0.293i)4-s + (0.776 − 0.630i)5-s + (0.774 + 0.562i)6-s + 0.795·7-s + (−0.286 − 0.207i)8-s + (0.257 − 0.791i)9-s + (0.593 + 0.384i)10-s + (−0.262 − 0.808i)11-s + (−0.209 + 0.643i)12-s + (−0.314 + 0.969i)13-s + (0.173 + 0.535i)14-s + (0.348 − 1.30i)15-s + (0.0772 − 0.237i)16-s + (0.651 + 0.473i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 + 0.143i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.989 + 0.143i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.93712 - 0.211564i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.93712 - 0.211564i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.73 + 1.40i)T \) |
| 19 | \( 1 + (-0.809 - 0.587i)T \) |
good | 3 | \( 1 + (-1.89 + 1.37i)T + (0.927 - 2.85i)T^{2} \) |
| 7 | \( 1 - 2.10T + 7T^{2} \) |
| 11 | \( 1 + (0.871 + 2.68i)T + (-8.89 + 6.46i)T^{2} \) |
| 13 | \( 1 + (1.13 - 3.49i)T + (-10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (-2.68 - 1.95i)T + (5.25 + 16.1i)T^{2} \) |
| 23 | \( 1 + (-0.636 - 1.95i)T + (-18.6 + 13.5i)T^{2} \) |
| 29 | \( 1 + (-2.53 + 1.84i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (4.67 + 3.39i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (-2.58 + 7.96i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (-1.01 + 3.11i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + 7.97T + 43T^{2} \) |
| 47 | \( 1 + (6.27 - 4.55i)T + (14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (3.73 - 2.71i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (1.79 - 5.51i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (-0.312 - 0.961i)T + (-49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + (-7.18 - 5.21i)T + (20.7 + 63.7i)T^{2} \) |
| 71 | \( 1 + (-0.946 + 0.687i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (3.47 + 10.6i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (-4.59 + 3.34i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (-11.0 - 8.04i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + (-3.96 - 12.1i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (13.5 - 9.81i)T + (29.9 - 92.2i)T^{2} \) |
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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
−9.526311675188653051456316758060, −9.016302392975298278428225862244, −8.119031610010606321515083449268, −7.78157554038282586469883343451, −6.68675660956956222795044084011, −5.74163523688787226842615945498, −4.91039491368197554064727492159, −3.69964725439930186455232379222, −2.37844905275967319956049556801, −1.38880193637189718190535190683,
1.71940006658324542617209138410, 2.80672691440404395239122266783, 3.36596107138389003812486584521, 4.76063544651013524559872414440, 5.24095616417799624048207038244, 6.68042112818452252018947734930, 7.83218086366216701249537937629, 8.516754968397360113410204086349, 9.655649790849188254396248819201, 9.900012855025694606341071569327