| L(s) = 1 | − 3.10·3-s + 2.52·5-s + 7-s + 6.62·9-s + 3.62·11-s − 4.72·13-s − 7.83·15-s + 4.20·17-s − 7.10·19-s − 3.10·21-s + 0.578·23-s + 1.37·25-s − 11.2·27-s + 8.20·29-s + 5.04·31-s − 11.2·33-s + 2.52·35-s − 3.04·37-s + 14.6·39-s − 0.205·41-s + 4.78·43-s + 16.7·45-s + 6.20·47-s + 49-s − 13.0·51-s − 2·53-s + 9.15·55-s + ⋯ |
| L(s) = 1 | − 1.79·3-s + 1.12·5-s + 0.377·7-s + 2.20·9-s + 1.09·11-s − 1.31·13-s − 2.02·15-s + 1.01·17-s − 1.62·19-s − 0.677·21-s + 0.120·23-s + 0.274·25-s − 2.16·27-s + 1.52·29-s + 0.906·31-s − 1.95·33-s + 0.426·35-s − 0.501·37-s + 2.35·39-s − 0.0321·41-s + 0.729·43-s + 2.49·45-s + 0.905·47-s + 0.142·49-s − 1.82·51-s − 0.274·53-s + 1.23·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.130493417\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.130493417\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 3 | \( 1 + 3.10T + 3T^{2} \) |
| 5 | \( 1 - 2.52T + 5T^{2} \) |
| 11 | \( 1 - 3.62T + 11T^{2} \) |
| 13 | \( 1 + 4.72T + 13T^{2} \) |
| 17 | \( 1 - 4.20T + 17T^{2} \) |
| 19 | \( 1 + 7.10T + 19T^{2} \) |
| 23 | \( 1 - 0.578T + 23T^{2} \) |
| 29 | \( 1 - 8.20T + 29T^{2} \) |
| 31 | \( 1 - 5.04T + 31T^{2} \) |
| 37 | \( 1 + 3.04T + 37T^{2} \) |
| 41 | \( 1 + 0.205T + 41T^{2} \) |
| 43 | \( 1 - 4.78T + 43T^{2} \) |
| 47 | \( 1 - 6.20T + 47T^{2} \) |
| 53 | \( 1 + 2T + 53T^{2} \) |
| 59 | \( 1 - 12.5T + 59T^{2} \) |
| 61 | \( 1 - 10.5T + 61T^{2} \) |
| 67 | \( 1 - 6.57T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 9.25T + 73T^{2} \) |
| 79 | \( 1 - 5.15T + 79T^{2} \) |
| 83 | \( 1 - 5.94T + 83T^{2} \) |
| 89 | \( 1 + 10.4T + 89T^{2} \) |
| 97 | \( 1 + 9.36T + 97T^{2} \) |
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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
−10.11841506893636532784139747210, −9.750643960602176668495015834455, −8.506333855216101386973108776204, −7.14089114238560939250357447676, −6.48353080845161311100452378975, −5.80187097083796055413108356331, −5.00592026109090745684312152051, −4.22414739826478400136047542284, −2.21615233262368572473735339389, −0.972986235916146502642456585416,
0.972986235916146502642456585416, 2.21615233262368572473735339389, 4.22414739826478400136047542284, 5.00592026109090745684312152051, 5.80187097083796055413108356331, 6.48353080845161311100452378975, 7.14089114238560939250357447676, 8.506333855216101386973108776204, 9.750643960602176668495015834455, 10.11841506893636532784139747210
