| L(s) = 1 | + 2.17·3-s + 0.630·5-s + 0.290·7-s + 1.70·9-s + 2.63·11-s + 4.41·13-s + 1.36·15-s − 7.49·17-s + 3.53·19-s + 0.630·21-s + 5.87·23-s − 4.60·25-s − 2.80·27-s − 0.369·29-s + 5.70·31-s + 5.70·33-s + 0.183·35-s + 3.44·37-s + 9.58·39-s − 2.18·41-s + 4.04·43-s + 1.07·45-s − 2.68·47-s − 6.91·49-s − 16.2·51-s + 53-s + 1.65·55-s + ⋯ |
| L(s) = 1 | + 1.25·3-s + 0.282·5-s + 0.109·7-s + 0.569·9-s + 0.793·11-s + 1.22·13-s + 0.353·15-s − 1.81·17-s + 0.811·19-s + 0.137·21-s + 1.22·23-s − 0.920·25-s − 0.539·27-s − 0.0685·29-s + 1.02·31-s + 0.993·33-s + 0.0310·35-s + 0.566·37-s + 1.53·39-s − 0.340·41-s + 0.617·43-s + 0.160·45-s − 0.390·47-s − 0.987·49-s − 2.27·51-s + 0.137·53-s + 0.223·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 848 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 848 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.607098436\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.607098436\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 53 | \( 1 - T \) |
| good | 3 | \( 1 - 2.17T + 3T^{2} \) |
| 5 | \( 1 - 0.630T + 5T^{2} \) |
| 7 | \( 1 - 0.290T + 7T^{2} \) |
| 11 | \( 1 - 2.63T + 11T^{2} \) |
| 13 | \( 1 - 4.41T + 13T^{2} \) |
| 17 | \( 1 + 7.49T + 17T^{2} \) |
| 19 | \( 1 - 3.53T + 19T^{2} \) |
| 23 | \( 1 - 5.87T + 23T^{2} \) |
| 29 | \( 1 + 0.369T + 29T^{2} \) |
| 31 | \( 1 - 5.70T + 31T^{2} \) |
| 37 | \( 1 - 3.44T + 37T^{2} \) |
| 41 | \( 1 + 2.18T + 41T^{2} \) |
| 43 | \( 1 - 4.04T + 43T^{2} \) |
| 47 | \( 1 + 2.68T + 47T^{2} \) |
| 59 | \( 1 - 3.07T + 59T^{2} \) |
| 61 | \( 1 + 5.12T + 61T^{2} \) |
| 67 | \( 1 + 8.94T + 67T^{2} \) |
| 71 | \( 1 + 4.85T + 71T^{2} \) |
| 73 | \( 1 - 8.63T + 73T^{2} \) |
| 79 | \( 1 + 10.0T + 79T^{2} \) |
| 83 | \( 1 + 4.64T + 83T^{2} \) |
| 89 | \( 1 + 1.31T + 89T^{2} \) |
| 97 | \( 1 - 7.04T + 97T^{2} \) |
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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
−9.927353540749727735474943638568, −9.067023321353784188936856614701, −8.741844393110921065049407289654, −7.82360678466619833895765384951, −6.79348197058635571697259845190, −5.97848541033109891184954164300, −4.57717332321279856294358161587, −3.64972041490207412070507744185, −2.67124643654544191792241349263, −1.48672371137600221039691326103,
1.48672371137600221039691326103, 2.67124643654544191792241349263, 3.64972041490207412070507744185, 4.57717332321279856294358161587, 5.97848541033109891184954164300, 6.79348197058635571697259845190, 7.82360678466619833895765384951, 8.741844393110921065049407289654, 9.067023321353784188936856614701, 9.927353540749727735474943638568
