| L(s) = 1 | + 2.76·2-s − 3-s + 5.65·4-s − 3.65·5-s − 2.76·6-s − 7-s + 10.1·8-s + 9-s − 10.1·10-s + 0.555·11-s − 5.65·12-s + 1.94·13-s − 2.76·14-s + 3.65·15-s + 16.6·16-s + 6.99·17-s + 2.76·18-s − 2.93·19-s − 20.6·20-s + 21-s + 1.53·22-s − 4.18·23-s − 10.1·24-s + 8.35·25-s + 5.39·26-s − 27-s − 5.65·28-s + ⋯ |
| L(s) = 1 | + 1.95·2-s − 0.577·3-s + 2.82·4-s − 1.63·5-s − 1.12·6-s − 0.377·7-s + 3.57·8-s + 0.333·9-s − 3.19·10-s + 0.167·11-s − 1.63·12-s + 0.540·13-s − 0.739·14-s + 0.943·15-s + 4.17·16-s + 1.69·17-s + 0.652·18-s − 0.673·19-s − 4.62·20-s + 0.218·21-s + 0.327·22-s − 0.872·23-s − 2.06·24-s + 1.67·25-s + 1.05·26-s − 0.192·27-s − 1.06·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.446662779\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.446662779\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 127 | \( 1 - T \) |
| good | 2 | \( 1 - 2.76T + 2T^{2} \) |
| 5 | \( 1 + 3.65T + 5T^{2} \) |
| 11 | \( 1 - 0.555T + 11T^{2} \) |
| 13 | \( 1 - 1.94T + 13T^{2} \) |
| 17 | \( 1 - 6.99T + 17T^{2} \) |
| 19 | \( 1 + 2.93T + 19T^{2} \) |
| 23 | \( 1 + 4.18T + 23T^{2} \) |
| 29 | \( 1 - 8.74T + 29T^{2} \) |
| 31 | \( 1 - 0.936T + 31T^{2} \) |
| 37 | \( 1 - 2.96T + 37T^{2} \) |
| 41 | \( 1 + 0.248T + 41T^{2} \) |
| 43 | \( 1 - 0.630T + 43T^{2} \) |
| 47 | \( 1 - 4.41T + 47T^{2} \) |
| 53 | \( 1 - 7.79T + 53T^{2} \) |
| 59 | \( 1 - 3.50T + 59T^{2} \) |
| 61 | \( 1 - 0.0482T + 61T^{2} \) |
| 67 | \( 1 - 1.92T + 67T^{2} \) |
| 71 | \( 1 + 6.19T + 71T^{2} \) |
| 73 | \( 1 - 12.2T + 73T^{2} \) |
| 79 | \( 1 + 15.7T + 79T^{2} \) |
| 83 | \( 1 + 13.8T + 83T^{2} \) |
| 89 | \( 1 - 4.29T + 89T^{2} \) |
| 97 | \( 1 - 12.7T + 97T^{2} \) |
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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
−8.418727448929603172547825568154, −7.73815342098560460746903492541, −7.07270263037910982405335083482, −6.31834632804019693134542826536, −5.68130157354715309214440614078, −4.75696967997747476786715555588, −4.04503058333913070702458417048, −3.58883340518180200573077202212, −2.68298209898678164988171301014, −1.07158948589009756934814586603,
1.07158948589009756934814586603, 2.68298209898678164988171301014, 3.58883340518180200573077202212, 4.04503058333913070702458417048, 4.75696967997747476786715555588, 5.68130157354715309214440614078, 6.31834632804019693134542826536, 7.07270263037910982405335083482, 7.73815342098560460746903492541, 8.418727448929603172547825568154
