L(s) = 1 | + 5-s + 1.56·7-s + 2·11-s + 0.561·13-s − 5.56·17-s − 2·19-s + 23-s + 25-s − 0.123·29-s − 8.12·31-s + 1.56·35-s − 3.56·37-s + 4.12·41-s − 10.2·43-s − 3.68·47-s − 4.56·49-s − 4.43·53-s + 2·55-s + 5.56·59-s − 9.12·61-s + 0.561·65-s − 11.5·67-s + 5·71-s + 3.43·73-s + 3.12·77-s − 9.12·79-s + 4.68·83-s + ⋯ |
L(s) = 1 | + 0.447·5-s + 0.590·7-s + 0.603·11-s + 0.155·13-s − 1.34·17-s − 0.458·19-s + 0.208·23-s + 0.200·25-s − 0.0228·29-s − 1.45·31-s + 0.263·35-s − 0.585·37-s + 0.643·41-s − 1.56·43-s − 0.537·47-s − 0.651·49-s − 0.609·53-s + 0.269·55-s + 0.724·59-s − 1.16·61-s + 0.0696·65-s − 1.41·67-s + 0.593·71-s + 0.402·73-s + 0.355·77-s − 1.02·79-s + 0.514·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 23 | \( 1 - T \) |
good | 7 | \( 1 - 1.56T + 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 - 0.561T + 13T^{2} \) |
| 17 | \( 1 + 5.56T + 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 29 | \( 1 + 0.123T + 29T^{2} \) |
| 31 | \( 1 + 8.12T + 31T^{2} \) |
| 37 | \( 1 + 3.56T + 37T^{2} \) |
| 41 | \( 1 - 4.12T + 41T^{2} \) |
| 43 | \( 1 + 10.2T + 43T^{2} \) |
| 47 | \( 1 + 3.68T + 47T^{2} \) |
| 53 | \( 1 + 4.43T + 53T^{2} \) |
| 59 | \( 1 - 5.56T + 59T^{2} \) |
| 61 | \( 1 + 9.12T + 61T^{2} \) |
| 67 | \( 1 + 11.5T + 67T^{2} \) |
| 71 | \( 1 - 5T + 71T^{2} \) |
| 73 | \( 1 - 3.43T + 73T^{2} \) |
| 79 | \( 1 + 9.12T + 79T^{2} \) |
| 83 | \( 1 - 4.68T + 83T^{2} \) |
| 89 | \( 1 + 8T + 89T^{2} \) |
| 97 | \( 1 + 3.12T + 97T^{2} \) |
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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
−7.39422581994701323855381367727, −6.70345532552626087413801820057, −6.19679791740728112859349488778, −5.31668119883247553272483166669, −4.67648906820745712549583430050, −3.97155902444509818503704612250, −3.07599016680113969757096304514, −2.01740960175750564163160153787, −1.49416692446185920011985731539, 0,
1.49416692446185920011985731539, 2.01740960175750564163160153787, 3.07599016680113969757096304514, 3.97155902444509818503704612250, 4.67648906820745712549583430050, 5.31668119883247553272483166669, 6.19679791740728112859349488778, 6.70345532552626087413801820057, 7.39422581994701323855381367727