L(s) = 1 | + (0.951 − 0.309i)2-s + (0.587 + 0.809i)3-s + (0.809 − 0.587i)4-s + (0.809 + 0.587i)6-s − 0.381i·7-s + (0.587 − 0.809i)8-s + (−0.309 + 0.951i)9-s + (0.427 + 1.31i)11-s + (0.951 + 0.309i)12-s + (2.35 + 0.763i)13-s + (−0.118 − 0.363i)14-s + (0.309 − 0.951i)16-s + (1.90 − 2.61i)17-s + i·18-s + (6.23 + 4.53i)19-s + ⋯ |
L(s) = 1 | + (0.672 − 0.218i)2-s + (0.339 + 0.467i)3-s + (0.404 − 0.293i)4-s + (0.330 + 0.239i)6-s − 0.144i·7-s + (0.207 − 0.286i)8-s + (−0.103 + 0.317i)9-s + (0.128 + 0.396i)11-s + (0.274 + 0.0892i)12-s + (0.652 + 0.211i)13-s + (−0.0315 − 0.0970i)14-s + (0.0772 − 0.237i)16-s + (0.461 − 0.634i)17-s + 0.235i·18-s + (1.43 + 1.03i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 - 0.137i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.990 - 0.137i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.67088 + 0.184014i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.67088 + 0.184014i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.951 + 0.309i)T \) |
| 3 | \( 1 + (-0.587 - 0.809i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 0.381iT - 7T^{2} \) |
| 11 | \( 1 + (-0.427 - 1.31i)T + (-8.89 + 6.46i)T^{2} \) |
| 13 | \( 1 + (-2.35 - 0.763i)T + (10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (-1.90 + 2.61i)T + (-5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (-6.23 - 4.53i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (-4.25 + 1.38i)T + (18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (0.381 - 0.277i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (3.54 + 2.57i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (7.60 + 2.47i)T + (29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (2.38 - 7.33i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 - 5.70iT - 43T^{2} \) |
| 47 | \( 1 + (6.88 + 9.47i)T + (-14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (5.34 + 7.35i)T + (-16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (-0.427 + 1.31i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (-2.23 - 6.88i)T + (-49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + (-6.15 + 8.47i)T + (-20.7 - 63.7i)T^{2} \) |
| 71 | \( 1 + (11.7 - 8.50i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (11.8 - 3.85i)T + (59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (2.73 - 1.98i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (-5.20 + 7.16i)T + (-25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + (3.85 + 11.8i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (3.30 + 4.54i)T + (-29.9 + 92.2i)T^{2} \) |
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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
−10.29797769272523827227563206256, −9.719238795567854377622404394035, −8.779284505227268091160620479639, −7.69408262095859836300088782121, −6.86391621928541989130607394755, −5.66456293690212771306016596477, −4.90218476368311844525245170265, −3.78673811430604561159871679564, −3.04817502601706717132384329775, −1.53040198591719094127270786439,
1.37431738993453741921826017995, 2.93839437813381365015575037790, 3.65710525085484438595134421191, 5.07905318581829259605726904189, 5.83473293632158239728638171832, 6.87692796339846026301936324728, 7.55473204255642647413441923357, 8.577557229054366373181661945610, 9.245479655372388881869952976176, 10.52651802573927920209877757725