L(s) = 1 | − 2.73·2-s + 3-s + 5.46·4-s − 5-s − 2.73·6-s − 9.46·8-s + 9-s + 2.73·10-s + 0.732·11-s + 5.46·12-s − 2.26·13-s − 15-s + 14.9·16-s − 3.26·17-s − 2.73·18-s − 4.46·19-s − 5.46·20-s − 2·22-s − 4.73·23-s − 9.46·24-s + 25-s + 6.19·26-s + 27-s − 4.19·29-s + 2.73·30-s + 0.464·31-s − 21.8·32-s + ⋯ |
L(s) = 1 | − 1.93·2-s + 0.577·3-s + 2.73·4-s − 0.447·5-s − 1.11·6-s − 3.34·8-s + 0.333·9-s + 0.863·10-s + 0.220·11-s + 1.57·12-s − 0.629·13-s − 0.258·15-s + 3.73·16-s − 0.792·17-s − 0.643·18-s − 1.02·19-s − 1.22·20-s − 0.426·22-s − 0.986·23-s − 1.93·24-s + 0.200·25-s + 1.21·26-s + 0.192·27-s − 0.779·29-s + 0.498·30-s + 0.0833·31-s − 3.86·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + 2.73T + 2T^{2} \) |
| 11 | \( 1 - 0.732T + 11T^{2} \) |
| 13 | \( 1 + 2.26T + 13T^{2} \) |
| 17 | \( 1 + 3.26T + 17T^{2} \) |
| 19 | \( 1 + 4.46T + 19T^{2} \) |
| 23 | \( 1 + 4.73T + 23T^{2} \) |
| 29 | \( 1 + 4.19T + 29T^{2} \) |
| 31 | \( 1 - 0.464T + 31T^{2} \) |
| 37 | \( 1 + 3.19T + 37T^{2} \) |
| 41 | \( 1 - 0.732T + 41T^{2} \) |
| 43 | \( 1 - 3.19T + 43T^{2} \) |
| 47 | \( 1 + 2T + 47T^{2} \) |
| 53 | \( 1 - 12.3T + 53T^{2} \) |
| 59 | \( 1 - 0.196T + 59T^{2} \) |
| 61 | \( 1 + 4T + 61T^{2} \) |
| 67 | \( 1 + 14.6T + 67T^{2} \) |
| 71 | \( 1 - 6.19T + 71T^{2} \) |
| 73 | \( 1 + 12.6T + 73T^{2} \) |
| 79 | \( 1 + 7.39T + 79T^{2} \) |
| 83 | \( 1 + 15.1T + 83T^{2} \) |
| 89 | \( 1 + 15.1T + 89T^{2} \) |
| 97 | \( 1 + 14.9T + 97T^{2} \) |
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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
−9.849543164466496226958806702435, −8.906914209968855158380828581096, −8.494829620778494367034299481028, −7.55163285068307729115697102553, −6.98658786520566803110093170981, −5.96228955158126698577274887989, −4.13747278538977431008876417058, −2.73395600229801674050154264895, −1.77352868017345184102614907117, 0,
1.77352868017345184102614907117, 2.73395600229801674050154264895, 4.13747278538977431008876417058, 5.96228955158126698577274887989, 6.98658786520566803110093170981, 7.55163285068307729115697102553, 8.494829620778494367034299481028, 8.906914209968855158380828581096, 9.849543164466496226958806702435