L(s) = 1 | − 1.50i·2-s + (−1.53 − 0.796i)3-s − 0.267·4-s + (1.12 − 1.93i)5-s + (−1.19 + 2.31i)6-s − 2.12·7-s − 2.60i·8-s + (1.73 + 2.44i)9-s + (−2.90 − 1.69i)10-s + (−3.27 − 0.517i)11-s + (0.412 + 0.213i)12-s + 2.12·13-s + 3.20i·14-s + (−3.27 + 2.07i)15-s − 4.46·16-s + 4.11i·17-s + ⋯ |
L(s) = 1 | − 1.06i·2-s + (−0.888 − 0.459i)3-s − 0.133·4-s + (0.503 − 0.863i)5-s + (−0.489 + 0.945i)6-s − 0.804·7-s − 0.922i·8-s + (0.577 + 0.816i)9-s + (−0.920 − 0.536i)10-s + (−0.987 − 0.156i)11-s + (0.118 + 0.0615i)12-s + 0.590·13-s + 0.857i·14-s + (−0.844 + 0.535i)15-s − 1.11·16-s + 0.997i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.917 + 0.397i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.917 + 0.397i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.183477 - 0.885349i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.183477 - 0.885349i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.53 + 0.796i)T \) |
| 5 | \( 1 + (-1.12 + 1.93i)T \) |
| 11 | \( 1 + (3.27 + 0.517i)T \) |
good | 2 | \( 1 + 1.50iT - 2T^{2} \) |
| 7 | \( 1 + 2.12T + 7T^{2} \) |
| 13 | \( 1 - 2.12T + 13T^{2} \) |
| 17 | \( 1 - 4.11iT - 17T^{2} \) |
| 19 | \( 1 + 4.63iT - 19T^{2} \) |
| 23 | \( 1 - 5.32T + 23T^{2} \) |
| 29 | \( 1 - 8.95T + 29T^{2} \) |
| 31 | \( 1 - 3.46T + 31T^{2} \) |
| 37 | \( 1 - 1.16iT - 37T^{2} \) |
| 41 | \( 1 - 2.39T + 41T^{2} \) |
| 43 | \( 1 - 9.50T + 43T^{2} \) |
| 47 | \( 1 + 3.07T + 47T^{2} \) |
| 53 | \( 1 + 4.50T + 53T^{2} \) |
| 59 | \( 1 + 4.89iT - 59T^{2} \) |
| 61 | \( 1 - 3.39iT - 61T^{2} \) |
| 67 | \( 1 - 12.3iT - 67T^{2} \) |
| 71 | \( 1 - 13.3iT - 71T^{2} \) |
| 73 | \( 1 + 2.12T + 73T^{2} \) |
| 79 | \( 1 - 13.8iT - 79T^{2} \) |
| 83 | \( 1 + 9.33iT - 83T^{2} \) |
| 89 | \( 1 + 4.62iT - 89T^{2} \) |
| 97 | \( 1 + 1.16iT - 97T^{2} \) |
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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
−12.60288892416910281230056103224, −11.38641025133518425569059526769, −10.57478245076603318527660043085, −9.782071806597621940828433243615, −8.464646077704556845388819396108, −6.85894132010235689444875897800, −5.90153568903307848495402997264, −4.56094901043770993451836954972, −2.68907604924610659289550643512, −0.998470260566059661271444834021,
2.97893233812162270888722508714, 4.97443825244924781801631314868, 6.04585979869464195957725485934, 6.64455256023483084735829277897, 7.70093901197317874586161931693, 9.308679251402325350822745874039, 10.35889145457063180559837853203, 11.03663065953562295654449260462, 12.23253876226632563497634740242, 13.46293034859606547367540682530