| L(s) = 1 | − 2.46·3-s − 2.10·5-s + 3.09·9-s − 4.68·11-s − 5.97·13-s + 5.20·15-s − 1.75·17-s − 3.07·19-s + 7.82·23-s − 0.561·25-s − 0.237·27-s − 8.57·29-s − 10.7·31-s + 11.5·33-s − 7.27·37-s + 14.7·39-s − 0.134·41-s − 2.06·43-s − 6.52·45-s − 2.55·47-s + 4.32·51-s − 6.06·53-s + 9.86·55-s + 7.59·57-s − 3.87·59-s + 3.04·61-s + 12.5·65-s + ⋯ |
| L(s) = 1 | − 1.42·3-s − 0.942·5-s + 1.03·9-s − 1.41·11-s − 1.65·13-s + 1.34·15-s − 0.424·17-s − 0.705·19-s + 1.63·23-s − 0.112·25-s − 0.0456·27-s − 1.59·29-s − 1.92·31-s + 2.01·33-s − 1.19·37-s + 2.36·39-s − 0.0210·41-s − 0.314·43-s − 0.972·45-s − 0.372·47-s + 0.605·51-s − 0.832·53-s + 1.33·55-s + 1.00·57-s − 0.504·59-s + 0.390·61-s + 1.56·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.04966250535\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.04966250535\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 2.46T + 3T^{2} \) |
| 5 | \( 1 + 2.10T + 5T^{2} \) |
| 11 | \( 1 + 4.68T + 11T^{2} \) |
| 13 | \( 1 + 5.97T + 13T^{2} \) |
| 17 | \( 1 + 1.75T + 17T^{2} \) |
| 19 | \( 1 + 3.07T + 19T^{2} \) |
| 23 | \( 1 - 7.82T + 23T^{2} \) |
| 29 | \( 1 + 8.57T + 29T^{2} \) |
| 31 | \( 1 + 10.7T + 31T^{2} \) |
| 37 | \( 1 + 7.27T + 37T^{2} \) |
| 41 | \( 1 + 0.134T + 41T^{2} \) |
| 43 | \( 1 + 2.06T + 43T^{2} \) |
| 47 | \( 1 + 2.55T + 47T^{2} \) |
| 53 | \( 1 + 6.06T + 53T^{2} \) |
| 59 | \( 1 + 3.87T + 59T^{2} \) |
| 61 | \( 1 - 3.04T + 61T^{2} \) |
| 67 | \( 1 - 5.06T + 67T^{2} \) |
| 71 | \( 1 + 5.33T + 71T^{2} \) |
| 73 | \( 1 + 11.4T + 73T^{2} \) |
| 79 | \( 1 - 15.7T + 79T^{2} \) |
| 83 | \( 1 - 8.25T + 83T^{2} \) |
| 89 | \( 1 + 4.46T + 89T^{2} \) |
| 97 | \( 1 - 0.302T + 97T^{2} \) |
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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
−8.834959377598580372410479722941, −7.72781551960465540711543123386, −7.32659607069614218876084204250, −6.62750044585225035648191262029, −5.33426020181692832080277885357, −5.23073802546187716949631946200, −4.32892065106771391679891395399, −3.20251439304380132467094031426, −2.01807283193730835904367408422, −0.13884148271989064950534601263,
0.13884148271989064950534601263, 2.01807283193730835904367408422, 3.20251439304380132467094031426, 4.32892065106771391679891395399, 5.23073802546187716949631946200, 5.33426020181692832080277885357, 6.62750044585225035648191262029, 7.32659607069614218876084204250, 7.72781551960465540711543123386, 8.834959377598580372410479722941
