| L(s) = 1 | − 2-s + 0.626·3-s + 4-s − 4.16·5-s − 0.626·6-s + 4.47·7-s − 8-s − 2.60·9-s + 4.16·10-s − 11-s + 0.626·12-s − 4.07·13-s − 4.47·14-s − 2.61·15-s + 16-s + 0.444·17-s + 2.60·18-s − 4.25·19-s − 4.16·20-s + 2.80·21-s + 22-s − 7.47·23-s − 0.626·24-s + 12.3·25-s + 4.07·26-s − 3.51·27-s + 4.47·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.361·3-s + 0.5·4-s − 1.86·5-s − 0.255·6-s + 1.69·7-s − 0.353·8-s − 0.869·9-s + 1.31·10-s − 0.301·11-s + 0.180·12-s − 1.12·13-s − 1.19·14-s − 0.674·15-s + 0.250·16-s + 0.107·17-s + 0.614·18-s − 0.976·19-s − 0.931·20-s + 0.611·21-s + 0.213·22-s − 1.55·23-s − 0.127·24-s + 2.47·25-s + 0.798·26-s − 0.676·27-s + 0.845·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6948458962\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6948458962\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 197 | \( 1 - T \) |
| good | 3 | \( 1 - 0.626T + 3T^{2} \) |
| 5 | \( 1 + 4.16T + 5T^{2} \) |
| 7 | \( 1 - 4.47T + 7T^{2} \) |
| 13 | \( 1 + 4.07T + 13T^{2} \) |
| 17 | \( 1 - 0.444T + 17T^{2} \) |
| 19 | \( 1 + 4.25T + 19T^{2} \) |
| 23 | \( 1 + 7.47T + 23T^{2} \) |
| 29 | \( 1 - 4.69T + 29T^{2} \) |
| 31 | \( 1 - 4.49T + 31T^{2} \) |
| 37 | \( 1 + 2.54T + 37T^{2} \) |
| 41 | \( 1 - 4.86T + 41T^{2} \) |
| 43 | \( 1 + 3.43T + 43T^{2} \) |
| 47 | \( 1 + 10.9T + 47T^{2} \) |
| 53 | \( 1 + 9.00T + 53T^{2} \) |
| 59 | \( 1 - 5.40T + 59T^{2} \) |
| 61 | \( 1 - 10.0T + 61T^{2} \) |
| 67 | \( 1 - 11.9T + 67T^{2} \) |
| 71 | \( 1 - 6.77T + 71T^{2} \) |
| 73 | \( 1 + 7.28T + 73T^{2} \) |
| 79 | \( 1 - 6.11T + 79T^{2} \) |
| 83 | \( 1 + 2.21T + 83T^{2} \) |
| 89 | \( 1 + 5.18T + 89T^{2} \) |
| 97 | \( 1 - 7.86T + 97T^{2} \) |
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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
−8.214892436326873134677164099582, −8.027617852825447372130123244118, −7.33591012240025168825110034011, −6.44636497352666774977301682115, −5.18825477770636033948208863517, −4.59901629035430881394552132655, −3.81955086954562953594485880677, −2.77282044889454808937299211935, −1.95845813413258244225640124423, −0.49281140127847001820994695371,
0.49281140127847001820994695371, 1.95845813413258244225640124423, 2.77282044889454808937299211935, 3.81955086954562953594485880677, 4.59901629035430881394552132655, 5.18825477770636033948208863517, 6.44636497352666774977301682115, 7.33591012240025168825110034011, 8.027617852825447372130123244118, 8.214892436326873134677164099582
