L(s) = 1 | − 2.26·7-s − 2.54·11-s − 5.02·13-s + 5.72·17-s − 1.86·19-s + 5.39·23-s − 3·29-s + 9.38·31-s − 2.59·37-s + 3.96·41-s + 10.2·43-s + 1.07·47-s − 1.86·49-s + 10.7·53-s − 1.58·59-s + 0.869·61-s − 4.85·67-s − 1.86·71-s + 3.87·73-s + 5.76·77-s − 13.2·79-s − 14.7·83-s − 13.4·89-s + 11.3·91-s + 17.6·97-s − 5.32·101-s − 6.01·103-s + ⋯ |
L(s) = 1 | − 0.856·7-s − 0.768·11-s − 1.39·13-s + 1.38·17-s − 0.429·19-s + 1.12·23-s − 0.557·29-s + 1.68·31-s − 0.426·37-s + 0.619·41-s + 1.55·43-s + 0.156·47-s − 0.267·49-s + 1.47·53-s − 0.206·59-s + 0.111·61-s − 0.593·67-s − 0.221·71-s + 0.453·73-s + 0.657·77-s − 1.49·79-s − 1.61·83-s − 1.42·89-s + 1.19·91-s + 1.79·97-s − 0.529·101-s − 0.592·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 2.26T + 7T^{2} \) |
| 11 | \( 1 + 2.54T + 11T^{2} \) |
| 13 | \( 1 + 5.02T + 13T^{2} \) |
| 17 | \( 1 - 5.72T + 17T^{2} \) |
| 19 | \( 1 + 1.86T + 19T^{2} \) |
| 23 | \( 1 - 5.39T + 23T^{2} \) |
| 29 | \( 1 + 3T + 29T^{2} \) |
| 31 | \( 1 - 9.38T + 31T^{2} \) |
| 37 | \( 1 + 2.59T + 37T^{2} \) |
| 41 | \( 1 - 3.96T + 41T^{2} \) |
| 43 | \( 1 - 10.2T + 43T^{2} \) |
| 47 | \( 1 - 1.07T + 47T^{2} \) |
| 53 | \( 1 - 10.7T + 53T^{2} \) |
| 59 | \( 1 + 1.58T + 59T^{2} \) |
| 61 | \( 1 - 0.869T + 61T^{2} \) |
| 67 | \( 1 + 4.85T + 67T^{2} \) |
| 71 | \( 1 + 1.86T + 71T^{2} \) |
| 73 | \( 1 - 3.87T + 73T^{2} \) |
| 79 | \( 1 + 13.2T + 79T^{2} \) |
| 83 | \( 1 + 14.7T + 83T^{2} \) |
| 89 | \( 1 + 13.4T + 89T^{2} \) |
| 97 | \( 1 - 17.6T + 97T^{2} \) |
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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
−7.40446627056329135040579039071, −6.95462786897500960040365129758, −6.01314382036260459640905693946, −5.42974233086345344737944844991, −4.72808012585118174423156865571, −3.88112979007498937617870549647, −2.82217552805318393691723962545, −2.60539910398028075536643206434, −1.12656824739779404097443394110, 0,
1.12656824739779404097443394110, 2.60539910398028075536643206434, 2.82217552805318393691723962545, 3.88112979007498937617870549647, 4.72808012585118174423156865571, 5.42974233086345344737944844991, 6.01314382036260459640905693946, 6.95462786897500960040365129758, 7.40446627056329135040579039071