| L(s) = 1 | − 1.35·3-s − 1.17·9-s + 3.15·11-s + 1.17·13-s + 2.07·17-s + 1.92·19-s − 4.51·23-s + 5.63·27-s − 6.97·29-s + 3.07·31-s − 4.25·33-s + 11.4·37-s − 1.58·39-s − 2.79·41-s + 3.22·43-s − 1.89·47-s − 2.80·51-s + 6.64·53-s − 2.60·57-s − 7.93·59-s − 8.15·61-s + 8.63·67-s + 6.09·69-s − 2.84·71-s + 5.35·73-s + 15.5·79-s − 4.08·81-s + ⋯ |
| L(s) = 1 | − 0.779·3-s − 0.391·9-s + 0.950·11-s + 0.325·13-s + 0.503·17-s + 0.442·19-s − 0.941·23-s + 1.08·27-s − 1.29·29-s + 0.552·31-s − 0.741·33-s + 1.88·37-s − 0.253·39-s − 0.436·41-s + 0.492·43-s − 0.276·47-s − 0.392·51-s + 0.913·53-s − 0.344·57-s − 1.03·59-s − 1.04·61-s + 1.05·67-s + 0.733·69-s − 0.337·71-s + 0.626·73-s + 1.74·79-s − 0.454·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.441519140\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.441519140\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 1.35T + 3T^{2} \) |
| 11 | \( 1 - 3.15T + 11T^{2} \) |
| 13 | \( 1 - 1.17T + 13T^{2} \) |
| 17 | \( 1 - 2.07T + 17T^{2} \) |
| 19 | \( 1 - 1.92T + 19T^{2} \) |
| 23 | \( 1 + 4.51T + 23T^{2} \) |
| 29 | \( 1 + 6.97T + 29T^{2} \) |
| 31 | \( 1 - 3.07T + 31T^{2} \) |
| 37 | \( 1 - 11.4T + 37T^{2} \) |
| 41 | \( 1 + 2.79T + 41T^{2} \) |
| 43 | \( 1 - 3.22T + 43T^{2} \) |
| 47 | \( 1 + 1.89T + 47T^{2} \) |
| 53 | \( 1 - 6.64T + 53T^{2} \) |
| 59 | \( 1 + 7.93T + 59T^{2} \) |
| 61 | \( 1 + 8.15T + 61T^{2} \) |
| 67 | \( 1 - 8.63T + 67T^{2} \) |
| 71 | \( 1 + 2.84T + 71T^{2} \) |
| 73 | \( 1 - 5.35T + 73T^{2} \) |
| 79 | \( 1 - 15.5T + 79T^{2} \) |
| 83 | \( 1 + 18.0T + 83T^{2} \) |
| 89 | \( 1 - 2.01T + 89T^{2} \) |
| 97 | \( 1 - 7.00T + 97T^{2} \) |
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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
−7.67990967928971198322362224806, −6.85903130818473300783372174574, −6.09620193611318941628745570311, −5.84706753306290008132571116632, −5.01026798041760941594267123074, −4.20120436874048327716419994161, −3.53373479206967720102692724041, −2.61518865487290648872126042396, −1.52951187350326269658854022103, −0.62337184204826408771196974194,
0.62337184204826408771196974194, 1.52951187350326269658854022103, 2.61518865487290648872126042396, 3.53373479206967720102692724041, 4.20120436874048327716419994161, 5.01026798041760941594267123074, 5.84706753306290008132571116632, 6.09620193611318941628745570311, 6.85903130818473300783372174574, 7.67990967928971198322362224806
