| L(s) = 1 | − 1.72·3-s − 2.22·5-s + 1.96·7-s − 0.0396·9-s − 11-s + 7.04·13-s + 3.83·15-s − 7.60·17-s + 19-s − 3.38·21-s + 6.60·23-s − 0.0396·25-s + 5.22·27-s − 6.25·29-s − 5.36·31-s + 1.72·33-s − 4.38·35-s + 7.61·37-s − 12.1·39-s − 9.40·41-s − 6.26·43-s + 0.0882·45-s + 10.4·47-s − 3.12·49-s + 13.0·51-s + 4.34·53-s + 2.22·55-s + ⋯ |
| L(s) = 1 | − 0.993·3-s − 0.996·5-s + 0.743·7-s − 0.0132·9-s − 0.301·11-s + 1.95·13-s + 0.989·15-s − 1.84·17-s + 0.229·19-s − 0.738·21-s + 1.37·23-s − 0.00792·25-s + 1.00·27-s − 1.16·29-s − 0.964·31-s + 0.299·33-s − 0.740·35-s + 1.25·37-s − 1.94·39-s − 1.46·41-s − 0.956·43-s + 0.0131·45-s + 1.52·47-s − 0.446·49-s + 1.83·51-s + 0.597·53-s + 0.300·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8852265026\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8852265026\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 1.72T + 3T^{2} \) |
| 5 | \( 1 + 2.22T + 5T^{2} \) |
| 7 | \( 1 - 1.96T + 7T^{2} \) |
| 13 | \( 1 - 7.04T + 13T^{2} \) |
| 17 | \( 1 + 7.60T + 17T^{2} \) |
| 23 | \( 1 - 6.60T + 23T^{2} \) |
| 29 | \( 1 + 6.25T + 29T^{2} \) |
| 31 | \( 1 + 5.36T + 31T^{2} \) |
| 37 | \( 1 - 7.61T + 37T^{2} \) |
| 41 | \( 1 + 9.40T + 41T^{2} \) |
| 43 | \( 1 + 6.26T + 43T^{2} \) |
| 47 | \( 1 - 10.4T + 47T^{2} \) |
| 53 | \( 1 - 4.34T + 53T^{2} \) |
| 59 | \( 1 + 2.10T + 59T^{2} \) |
| 61 | \( 1 + 8.77T + 61T^{2} \) |
| 67 | \( 1 - 13.7T + 67T^{2} \) |
| 71 | \( 1 + 1.14T + 71T^{2} \) |
| 73 | \( 1 + 6.04T + 73T^{2} \) |
| 79 | \( 1 - 5.68T + 79T^{2} \) |
| 83 | \( 1 + 9.02T + 83T^{2} \) |
| 89 | \( 1 + 4.83T + 89T^{2} \) |
| 97 | \( 1 - 12.0T + 97T^{2} \) |
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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
−8.702262216019214448572135183579, −7.88313883622852445120446030714, −7.07121434855324814294094164788, −6.34155605362548840512647123193, −5.58453421842054729860960360404, −4.81551851813934216116398829041, −4.07816011197735133719052690780, −3.24064191224085637943820577349, −1.81247507996602143011564296650, −0.59341200883244010662176331745,
0.59341200883244010662176331745, 1.81247507996602143011564296650, 3.24064191224085637943820577349, 4.07816011197735133719052690780, 4.81551851813934216116398829041, 5.58453421842054729860960360404, 6.34155605362548840512647123193, 7.07121434855324814294094164788, 7.88313883622852445120446030714, 8.702262216019214448572135183579
