| L(s) = 1 | − 0.183·2-s − 2.64·3-s − 1.96·4-s + 0.484·6-s − 4.36·7-s + 0.727·8-s + 3.97·9-s − 0.259·11-s + 5.19·12-s − 0.888·13-s + 0.800·14-s + 3.79·16-s − 3.65·17-s − 0.728·18-s + 3.95·19-s + 11.5·21-s + 0.0475·22-s + 2.56·23-s − 1.92·24-s + 0.162·26-s − 2.57·27-s + 8.57·28-s − 2.50·29-s − 6.76·31-s − 2.15·32-s + 0.684·33-s + 0.670·34-s + ⋯ |
| L(s) = 1 | − 0.129·2-s − 1.52·3-s − 0.983·4-s + 0.197·6-s − 1.64·7-s + 0.257·8-s + 1.32·9-s − 0.0781·11-s + 1.49·12-s − 0.246·13-s + 0.213·14-s + 0.949·16-s − 0.886·17-s − 0.171·18-s + 0.908·19-s + 2.51·21-s + 0.0101·22-s + 0.535·23-s − 0.392·24-s + 0.0319·26-s − 0.494·27-s + 1.62·28-s − 0.464·29-s − 1.21·31-s − 0.380·32-s + 0.119·33-s + 0.115·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.003668246022\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.003668246022\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 197 | \( 1 + T \) |
| good | 2 | \( 1 + 0.183T + 2T^{2} \) |
| 3 | \( 1 + 2.64T + 3T^{2} \) |
| 7 | \( 1 + 4.36T + 7T^{2} \) |
| 11 | \( 1 + 0.259T + 11T^{2} \) |
| 13 | \( 1 + 0.888T + 13T^{2} \) |
| 17 | \( 1 + 3.65T + 17T^{2} \) |
| 19 | \( 1 - 3.95T + 19T^{2} \) |
| 23 | \( 1 - 2.56T + 23T^{2} \) |
| 29 | \( 1 + 2.50T + 29T^{2} \) |
| 31 | \( 1 + 6.76T + 31T^{2} \) |
| 37 | \( 1 + 9.87T + 37T^{2} \) |
| 41 | \( 1 + 10.6T + 41T^{2} \) |
| 43 | \( 1 + 4.66T + 43T^{2} \) |
| 47 | \( 1 + 11.0T + 47T^{2} \) |
| 53 | \( 1 + 10.0T + 53T^{2} \) |
| 59 | \( 1 + 8.02T + 59T^{2} \) |
| 61 | \( 1 - 2.13T + 61T^{2} \) |
| 67 | \( 1 + 1.00T + 67T^{2} \) |
| 71 | \( 1 + 0.407T + 71T^{2} \) |
| 73 | \( 1 - 1.66T + 73T^{2} \) |
| 79 | \( 1 - 3.08T + 79T^{2} \) |
| 83 | \( 1 - 12.2T + 83T^{2} \) |
| 89 | \( 1 + 6.10T + 89T^{2} \) |
| 97 | \( 1 + 8.15T + 97T^{2} \) |
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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
−8.348811675403205095483634249536, −7.23777232816997929751415706619, −6.71287532696150428558809252081, −6.09167072915374353138101077746, −5.17942443444345990920534752610, −4.95271419842830442191274970541, −3.75094278071744066533552927877, −3.17884264527991712336692651702, −1.50106250291919630617288439099, −0.03780971162271424173118123128,
0.03780971162271424173118123128, 1.50106250291919630617288439099, 3.17884264527991712336692651702, 3.75094278071744066533552927877, 4.95271419842830442191274970541, 5.17942443444345990920534752610, 6.09167072915374353138101077746, 6.71287532696150428558809252081, 7.23777232816997929751415706619, 8.348811675403205095483634249536
