| L(s) = 1 | − 1.82·2-s − 3-s + 1.34·4-s − 1.45·5-s + 1.82·6-s − 7-s + 1.19·8-s + 9-s + 2.66·10-s + 2.19·11-s − 1.34·12-s + 6.28·13-s + 1.82·14-s + 1.45·15-s − 4.87·16-s + 6.17·17-s − 1.82·18-s + 7.62·19-s − 1.95·20-s + 21-s − 4.00·22-s + 5.57·23-s − 1.19·24-s − 2.88·25-s − 11.5·26-s − 27-s − 1.34·28-s + ⋯ |
| L(s) = 1 | − 1.29·2-s − 0.577·3-s + 0.673·4-s − 0.650·5-s + 0.746·6-s − 0.377·7-s + 0.422·8-s + 0.333·9-s + 0.841·10-s + 0.660·11-s − 0.388·12-s + 1.74·13-s + 0.488·14-s + 0.375·15-s − 1.21·16-s + 1.49·17-s − 0.431·18-s + 1.74·19-s − 0.438·20-s + 0.218·21-s − 0.854·22-s + 1.16·23-s − 0.243·24-s − 0.576·25-s − 2.25·26-s − 0.192·27-s − 0.254·28-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
| \(L(1)\) |
\(\approx\) |
\(0.7976699142\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7976699142\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.82T + 2T^{2} \) |
| 5 | \( 1 + 1.45T + 5T^{2} \) |
| 11 | \( 1 - 2.19T + 11T^{2} \) |
| 13 | \( 1 - 6.28T + 13T^{2} \) |
| 17 | \( 1 - 6.17T + 17T^{2} \) |
| 19 | \( 1 - 7.62T + 19T^{2} \) |
| 23 | \( 1 - 5.57T + 23T^{2} \) |
| 29 | \( 1 - 3.99T + 29T^{2} \) |
| 31 | \( 1 + 3.97T + 31T^{2} \) |
| 37 | \( 1 + 1.95T + 37T^{2} \) |
| 41 | \( 1 - 9.92T + 41T^{2} \) |
| 43 | \( 1 + 7.53T + 43T^{2} \) |
| 47 | \( 1 + 1.82T + 47T^{2} \) |
| 53 | \( 1 + 2.70T + 53T^{2} \) |
| 59 | \( 1 + 5.00T + 59T^{2} \) |
| 61 | \( 1 - 4.52T + 61T^{2} \) |
| 67 | \( 1 - 4.03T + 67T^{2} \) |
| 71 | \( 1 - 13.6T + 71T^{2} \) |
| 73 | \( 1 + 0.355T + 73T^{2} \) |
| 79 | \( 1 + 5.46T + 79T^{2} \) |
| 83 | \( 1 + 0.444T + 83T^{2} \) |
| 89 | \( 1 + 6.92T + 89T^{2} \) |
| 97 | \( 1 - 2.38T + 97T^{2} \) |
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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
−8.906773689504493325055554951217, −8.118696896734778123287733791827, −7.50861905533346448080699588107, −6.80892402860434370956794859028, −5.93097119215377945134433041567, −5.06963036037562687836171426126, −3.87630527852590600559527029058, −3.25081065299899415337276829359, −1.37139661175835761379940719752, −0.831590608301711191694776718944,
0.831590608301711191694776718944, 1.37139661175835761379940719752, 3.25081065299899415337276829359, 3.87630527852590600559527029058, 5.06963036037562687836171426126, 5.93097119215377945134433041567, 6.80892402860434370956794859028, 7.50861905533346448080699588107, 8.118696896734778123287733791827, 8.906773689504493325055554951217
