| L(s) = 1 | + 0.772·2-s − 1.40·4-s − 2.17·7-s − 2.62·8-s − 3·11-s + 0.629·13-s − 1.68·14-s + 0.772·16-s + 4.17·17-s + 4.80·19-s − 2.31·22-s + 2.08·23-s + 0.486·26-s + 3.05·28-s + 29-s + 5.85·32-s + 3.22·34-s − 6.08·37-s + 3.71·38-s − 0.824·41-s + 8.72·43-s + 4.20·44-s + 1.61·46-s − 8.98·47-s − 2.26·49-s − 0.883·52-s + 6.88·53-s + ⋯ |
| L(s) = 1 | + 0.546·2-s − 0.701·4-s − 0.822·7-s − 0.929·8-s − 0.904·11-s + 0.174·13-s − 0.449·14-s + 0.193·16-s + 1.01·17-s + 1.10·19-s − 0.494·22-s + 0.434·23-s + 0.0954·26-s + 0.576·28-s + 0.185·29-s + 1.03·32-s + 0.553·34-s − 1.00·37-s + 0.602·38-s − 0.128·41-s + 1.32·43-s + 0.634·44-s + 0.237·46-s − 1.30·47-s − 0.323·49-s − 0.122·52-s + 0.946·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 - 0.772T + 2T^{2} \) |
| 7 | \( 1 + 2.17T + 7T^{2} \) |
| 11 | \( 1 + 3T + 11T^{2} \) |
| 13 | \( 1 - 0.629T + 13T^{2} \) |
| 17 | \( 1 - 4.17T + 17T^{2} \) |
| 19 | \( 1 - 4.80T + 19T^{2} \) |
| 23 | \( 1 - 2.08T + 23T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 6.08T + 37T^{2} \) |
| 41 | \( 1 + 0.824T + 41T^{2} \) |
| 43 | \( 1 - 8.72T + 43T^{2} \) |
| 47 | \( 1 + 8.98T + 47T^{2} \) |
| 53 | \( 1 - 6.88T + 53T^{2} \) |
| 59 | \( 1 + 6.45T + 59T^{2} \) |
| 61 | \( 1 + 2.80T + 61T^{2} \) |
| 67 | \( 1 - 11.0T + 67T^{2} \) |
| 71 | \( 1 + 2.63T + 71T^{2} \) |
| 73 | \( 1 - 14.7T + 73T^{2} \) |
| 79 | \( 1 + 12.0T + 79T^{2} \) |
| 83 | \( 1 + 7.62T + 83T^{2} \) |
| 89 | \( 1 + 17.8T + 89T^{2} \) |
| 97 | \( 1 + 0.538T + 97T^{2} \) |
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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
−7.67170309711443614939545958380, −6.90084854731077425834493180007, −6.03519209783934509962549554488, −5.39848257259793696382675504556, −4.95009326479529262835584847000, −3.90870552729687028089328655057, −3.26535314332877966635901021664, −2.70331110199360862416504296533, −1.15604607510574457959240121820, 0,
1.15604607510574457959240121820, 2.70331110199360862416504296533, 3.26535314332877966635901021664, 3.90870552729687028089328655057, 4.95009326479529262835584847000, 5.39848257259793696382675504556, 6.03519209783934509962549554488, 6.90084854731077425834493180007, 7.67170309711443614939545958380
