L(s) = 1 | + 5-s + 4.12·7-s + 4.64·11-s + 2.64·13-s + 4.12·17-s − 1.60·19-s − 23-s + 25-s − 1.48·29-s − 0.515·31-s + 4.12·35-s − 2.12·37-s + 4.51·41-s + 5.28·43-s + 2.39·47-s + 10.0·49-s + 4.12·53-s + 4.64·55-s − 3.79·59-s + 2.32·61-s + 2.64·65-s − 0.185·67-s + 14.9·71-s − 10.1·73-s + 19.1·77-s − 1.93·79-s + 0.844·83-s + ⋯ |
L(s) = 1 | + 0.447·5-s + 1.55·7-s + 1.39·11-s + 0.732·13-s + 1.00·17-s − 0.369·19-s − 0.208·23-s + 0.200·25-s − 0.275·29-s − 0.0925·31-s + 0.697·35-s − 0.349·37-s + 0.705·41-s + 0.805·43-s + 0.348·47-s + 1.43·49-s + 0.566·53-s + 0.625·55-s − 0.494·59-s + 0.298·61-s + 0.327·65-s − 0.0226·67-s + 1.77·71-s − 1.19·73-s + 2.18·77-s − 0.218·79-s + 0.0927·83-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
\(L(1)\) |
\(\approx\) |
\(3.586938278\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.586938278\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 4.12T + 7T^{2} \) |
| 11 | \( 1 - 4.64T + 11T^{2} \) |
| 13 | \( 1 - 2.64T + 13T^{2} \) |
| 17 | \( 1 - 4.12T + 17T^{2} \) |
| 19 | \( 1 + 1.60T + 19T^{2} \) |
| 29 | \( 1 + 1.48T + 29T^{2} \) |
| 31 | \( 1 + 0.515T + 31T^{2} \) |
| 37 | \( 1 + 2.12T + 37T^{2} \) |
| 41 | \( 1 - 4.51T + 41T^{2} \) |
| 43 | \( 1 - 5.28T + 43T^{2} \) |
| 47 | \( 1 - 2.39T + 47T^{2} \) |
| 53 | \( 1 - 4.12T + 53T^{2} \) |
| 59 | \( 1 + 3.79T + 59T^{2} \) |
| 61 | \( 1 - 2.32T + 61T^{2} \) |
| 67 | \( 1 + 0.185T + 67T^{2} \) |
| 71 | \( 1 - 14.9T + 71T^{2} \) |
| 73 | \( 1 + 10.1T + 73T^{2} \) |
| 79 | \( 1 + 1.93T + 79T^{2} \) |
| 83 | \( 1 - 0.844T + 83T^{2} \) |
| 89 | \( 1 + 0.780T + 89T^{2} \) |
| 97 | \( 1 + 9.21T + 97T^{2} \) |
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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
−7.86045931477227275667962074965, −7.17269483475898249163540497564, −6.34699984018479211499379168701, −5.72227842405176980325646295639, −5.07906816794478920043471446268, −4.18636207563779113946576057425, −3.72196090887798861878111956788, −2.50158112456511000198698090937, −1.56388609706987020206002817225, −1.08771509451387408035823179124,
1.08771509451387408035823179124, 1.56388609706987020206002817225, 2.50158112456511000198698090937, 3.72196090887798861878111956788, 4.18636207563779113946576057425, 5.07906816794478920043471446268, 5.72227842405176980325646295639, 6.34699984018479211499379168701, 7.17269483475898249163540497564, 7.86045931477227275667962074965