| L(s) = 1 | − 2.70·2-s − 2.87·3-s + 5.31·4-s + 4.01·5-s + 7.76·6-s + 2.72·7-s − 8.95·8-s + 5.24·9-s − 10.8·10-s + 4.57·11-s − 15.2·12-s + 5.43·13-s − 7.37·14-s − 11.5·15-s + 13.5·16-s + 6.82·17-s − 14.1·18-s − 8.08·19-s + 21.3·20-s − 7.83·21-s − 12.3·22-s − 5.44·23-s + 25.7·24-s + 11.1·25-s − 14.7·26-s − 6.45·27-s + 14.4·28-s + ⋯ |
| L(s) = 1 | − 1.91·2-s − 1.65·3-s + 2.65·4-s + 1.79·5-s + 3.17·6-s + 1.03·7-s − 3.16·8-s + 1.74·9-s − 3.43·10-s + 1.37·11-s − 4.40·12-s + 1.50·13-s − 1.97·14-s − 2.97·15-s + 3.39·16-s + 1.65·17-s − 3.34·18-s − 1.85·19-s + 4.77·20-s − 1.70·21-s − 2.63·22-s − 1.13·23-s + 5.24·24-s + 2.22·25-s − 2.88·26-s − 1.24·27-s + 2.73·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9280305705\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9280305705\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 4027 | \( 1+O(T) \) |
| good | 2 | \( 1 + 2.70T + 2T^{2} \) |
| 3 | \( 1 + 2.87T + 3T^{2} \) |
| 5 | \( 1 - 4.01T + 5T^{2} \) |
| 7 | \( 1 - 2.72T + 7T^{2} \) |
| 11 | \( 1 - 4.57T + 11T^{2} \) |
| 13 | \( 1 - 5.43T + 13T^{2} \) |
| 17 | \( 1 - 6.82T + 17T^{2} \) |
| 19 | \( 1 + 8.08T + 19T^{2} \) |
| 23 | \( 1 + 5.44T + 23T^{2} \) |
| 29 | \( 1 + 1.32T + 29T^{2} \) |
| 31 | \( 1 + 3.22T + 31T^{2} \) |
| 37 | \( 1 + 1.86T + 37T^{2} \) |
| 41 | \( 1 + 5.02T + 41T^{2} \) |
| 43 | \( 1 - 4.95T + 43T^{2} \) |
| 47 | \( 1 + 2.17T + 47T^{2} \) |
| 53 | \( 1 - 5.21T + 53T^{2} \) |
| 59 | \( 1 - 1.93T + 59T^{2} \) |
| 61 | \( 1 + 9.27T + 61T^{2} \) |
| 67 | \( 1 - 8.18T + 67T^{2} \) |
| 71 | \( 1 - 5.07T + 71T^{2} \) |
| 73 | \( 1 - 15.7T + 73T^{2} \) |
| 79 | \( 1 + 1.61T + 79T^{2} \) |
| 83 | \( 1 - 7.53T + 83T^{2} \) |
| 89 | \( 1 + 0.120T + 89T^{2} \) |
| 97 | \( 1 - 2.97T + 97T^{2} \) |
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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
−8.627950263609533211783097590340, −7.86381281620082088004519672010, −6.73779253622263909269809241673, −6.30312275880290026157899999444, −5.94348733568014988772599989528, −5.21749196202239793966113197987, −3.80626780357531964703146205892, −1.99830990469354899577117366624, −1.54970679626388507038600872727, −0.905710029527561541313654520854,
0.905710029527561541313654520854, 1.54970679626388507038600872727, 1.99830990469354899577117366624, 3.80626780357531964703146205892, 5.21749196202239793966113197987, 5.94348733568014988772599989528, 6.30312275880290026157899999444, 6.73779253622263909269809241673, 7.86381281620082088004519672010, 8.627950263609533211783097590340
