| L(s) = 1 | − 0.618·2-s + 0.381·3-s − 1.61·4-s − 0.236·6-s + 3.85·7-s + 2.23·8-s − 2.85·9-s + 11-s − 0.618·12-s + 1.76·13-s − 2.38·14-s + 1.85·16-s + 1.61·17-s + 1.76·18-s + 6.70·19-s + 1.47·21-s − 0.618·22-s + 7.09·23-s + 0.854·24-s − 1.09·26-s − 2.23·27-s − 6.23·28-s − 3.61·29-s − 3·31-s − 5.61·32-s + 0.381·33-s − 1.00·34-s + ⋯ |
| L(s) = 1 | − 0.437·2-s + 0.220·3-s − 0.809·4-s − 0.0963·6-s + 1.45·7-s + 0.790·8-s − 0.951·9-s + 0.301·11-s − 0.178·12-s + 0.489·13-s − 0.636·14-s + 0.463·16-s + 0.392·17-s + 0.415·18-s + 1.53·19-s + 0.321·21-s − 0.131·22-s + 1.47·23-s + 0.174·24-s − 0.213·26-s − 0.430·27-s − 1.17·28-s − 0.671·29-s − 0.538·31-s − 0.993·32-s + 0.0664·33-s − 0.171·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.062798124\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.062798124\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 + 0.618T + 2T^{2} \) |
| 3 | \( 1 - 0.381T + 3T^{2} \) |
| 7 | \( 1 - 3.85T + 7T^{2} \) |
| 13 | \( 1 - 1.76T + 13T^{2} \) |
| 17 | \( 1 - 1.61T + 17T^{2} \) |
| 19 | \( 1 - 6.70T + 19T^{2} \) |
| 23 | \( 1 - 7.09T + 23T^{2} \) |
| 29 | \( 1 + 3.61T + 29T^{2} \) |
| 31 | \( 1 + 3T + 31T^{2} \) |
| 37 | \( 1 - 5.76T + 37T^{2} \) |
| 41 | \( 1 + 3T + 41T^{2} \) |
| 43 | \( 1 + 6T + 43T^{2} \) |
| 47 | \( 1 + 5.94T + 47T^{2} \) |
| 53 | \( 1 + 6.32T + 53T^{2} \) |
| 59 | \( 1 - 9.47T + 59T^{2} \) |
| 61 | \( 1 + 11.0T + 61T^{2} \) |
| 67 | \( 1 - 8T + 67T^{2} \) |
| 71 | \( 1 + 14.1T + 71T^{2} \) |
| 73 | \( 1 - 12.6T + 73T^{2} \) |
| 79 | \( 1 + 0.854T + 79T^{2} \) |
| 83 | \( 1 + 16.8T + 83T^{2} \) |
| 89 | \( 1 + 18.0T + 89T^{2} \) |
| 97 | \( 1 + 0.618T + 97T^{2} \) |
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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
−11.52264203416942008975233426981, −11.13321331222576933286355994606, −9.783292997726884502134524247927, −8.898858358095854824851867634721, −8.215260952159068076810007580587, −7.38900284317193437643453877793, −5.57379634547833397679397177717, −4.80982157861880810380258122092, −3.36127819628620222821894641488, −1.34069691170114116461753547816,
1.34069691170114116461753547816, 3.36127819628620222821894641488, 4.80982157861880810380258122092, 5.57379634547833397679397177717, 7.38900284317193437643453877793, 8.215260952159068076810007580587, 8.898858358095854824851867634721, 9.783292997726884502134524247927, 11.13321331222576933286355994606, 11.52264203416942008975233426981
