| L(s) = 1 | − 1.66·2-s + 0.562·3-s + 0.774·4-s − 5-s − 0.936·6-s + 2.04·8-s − 2.68·9-s + 1.66·10-s + 11-s + 0.435·12-s + 1.60·13-s − 0.562·15-s − 4.94·16-s + 6.87·17-s + 4.47·18-s + 5.70·19-s − 0.774·20-s − 1.66·22-s + 1.89·23-s + 1.14·24-s + 25-s − 2.67·26-s − 3.19·27-s − 2.63·29-s + 0.936·30-s − 9.26·31-s + 4.16·32-s + ⋯ |
| L(s) = 1 | − 1.17·2-s + 0.324·3-s + 0.387·4-s − 0.447·5-s − 0.382·6-s + 0.721·8-s − 0.894·9-s + 0.526·10-s + 0.301·11-s + 0.125·12-s + 0.446·13-s − 0.145·15-s − 1.23·16-s + 1.66·17-s + 1.05·18-s + 1.30·19-s − 0.173·20-s − 0.355·22-s + 0.394·23-s + 0.234·24-s + 0.200·25-s − 0.525·26-s − 0.615·27-s − 0.489·29-s + 0.170·30-s − 1.66·31-s + 0.735·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2695 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2695 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8740427913\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8740427913\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 + 1.66T + 2T^{2} \) |
| 3 | \( 1 - 0.562T + 3T^{2} \) |
| 13 | \( 1 - 1.60T + 13T^{2} \) |
| 17 | \( 1 - 6.87T + 17T^{2} \) |
| 19 | \( 1 - 5.70T + 19T^{2} \) |
| 23 | \( 1 - 1.89T + 23T^{2} \) |
| 29 | \( 1 + 2.63T + 29T^{2} \) |
| 31 | \( 1 + 9.26T + 31T^{2} \) |
| 37 | \( 1 + 2.44T + 37T^{2} \) |
| 41 | \( 1 + 6.92T + 41T^{2} \) |
| 43 | \( 1 - 8.12T + 43T^{2} \) |
| 47 | \( 1 + 4.12T + 47T^{2} \) |
| 53 | \( 1 + 10.8T + 53T^{2} \) |
| 59 | \( 1 - 7.01T + 59T^{2} \) |
| 61 | \( 1 - 10.8T + 61T^{2} \) |
| 67 | \( 1 - 4.42T + 67T^{2} \) |
| 71 | \( 1 + 0.653T + 71T^{2} \) |
| 73 | \( 1 + 2.85T + 73T^{2} \) |
| 79 | \( 1 - 5.61T + 79T^{2} \) |
| 83 | \( 1 + 2.57T + 83T^{2} \) |
| 89 | \( 1 - 11.7T + 89T^{2} \) |
| 97 | \( 1 + 7.61T + 97T^{2} \) |
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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
−8.851722858569689090886701711941, −8.148031105245035168278501653254, −7.62113514861873500597209853340, −6.96957629224082950204751989490, −5.71839161819660133624475976372, −5.09749065334257750308382866218, −3.73936596598874004325953334036, −3.18405611350590126418885703618, −1.76452507517912621666083414930, −0.71027029110659724943248969101,
0.71027029110659724943248969101, 1.76452507517912621666083414930, 3.18405611350590126418885703618, 3.73936596598874004325953334036, 5.09749065334257750308382866218, 5.71839161819660133624475976372, 6.96957629224082950204751989490, 7.62113514861873500597209853340, 8.148031105245035168278501653254, 8.851722858569689090886701711941
