L(s) = 1 | + (0.129 + 0.224i)2-s + (−0.458 − 1.67i)3-s + (0.966 − 1.67i)4-s + 5-s + (0.316 − 0.319i)6-s + (2.52 − 0.775i)7-s + 1.02·8-s + (−2.57 + 1.53i)9-s + (0.129 + 0.224i)10-s − 1.45·11-s + (−3.23 − 0.846i)12-s + (−0.192 − 0.333i)13-s + (0.502 + 0.468i)14-s + (−0.458 − 1.67i)15-s + (−1.79 − 3.11i)16-s + (2.42 + 4.20i)17-s + ⋯ |
L(s) = 1 | + (0.0918 + 0.159i)2-s + (−0.264 − 0.964i)3-s + (0.483 − 0.836i)4-s + 0.447·5-s + (0.129 − 0.130i)6-s + (0.956 − 0.293i)7-s + 0.361·8-s + (−0.859 + 0.510i)9-s + (0.0410 + 0.0711i)10-s − 0.438·11-s + (−0.934 − 0.244i)12-s + (−0.0533 − 0.0923i)13-s + (0.134 + 0.125i)14-s + (−0.118 − 0.431i)15-s + (−0.449 − 0.779i)16-s + (0.588 + 1.01i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.197 + 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.197 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.20089 - 0.983354i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.20089 - 0.983354i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.458 + 1.67i)T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + (-2.52 + 0.775i)T \) |
good | 2 | \( 1 + (-0.129 - 0.224i)T + (-1 + 1.73i)T^{2} \) |
| 11 | \( 1 + 1.45T + 11T^{2} \) |
| 13 | \( 1 + (0.192 + 0.333i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-2.42 - 4.20i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.194 + 0.337i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + 7.95T + 23T^{2} \) |
| 29 | \( 1 + (-2.48 + 4.30i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-4.69 + 8.13i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (5.98 - 10.3i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-3.47 - 6.02i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (1.06 - 1.83i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-4.31 - 7.46i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-2.14 - 3.71i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (4.41 - 7.64i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (3.87 + 6.71i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.15 + 3.72i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 1.38T + 71T^{2} \) |
| 73 | \( 1 + (-5.49 - 9.52i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.839 - 1.45i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-6.11 + 10.5i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (3.23 - 5.61i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (0.759 - 1.31i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−11.50036374307360517201679271469, −10.58897817092737843107482993314, −9.873921894633015798373153903469, −8.189547991705523271478127969568, −7.67470894658096069764223654435, −6.33001362804112908683120743360, −5.81272101432594404177517312965, −4.64328352157569635251332163147, −2.37881225861349811055803229912, −1.29273891684812567378831029950,
2.27612078821231403504833876905, 3.56090417076188673585005416390, 4.80538590359847110914628259935, 5.68140036148672983448491103892, 7.09820580563805972259281164023, 8.211068112261786130723335029886, 9.019614992805220713828465911919, 10.27049852472552222237348555990, 10.86053216164193365020611318496, 11.98647003289222776465477454037