| L(s) = 1 | − 10.3·2-s + 4.60·3-s + 76.1·4-s + 25·5-s − 47.8·6-s + 17.5·7-s − 459.·8-s − 221.·9-s − 259.·10-s + 121·11-s + 350.·12-s − 1.17e3·13-s − 182.·14-s + 115.·15-s + 2.33e3·16-s + 523.·17-s + 2.30e3·18-s + 361·19-s + 1.90e3·20-s + 80.7·21-s − 1.25e3·22-s − 1.60e3·23-s − 2.11e3·24-s + 625·25-s + 1.22e4·26-s − 2.14e3·27-s + 1.33e3·28-s + ⋯ |
| L(s) = 1 | − 1.83·2-s + 0.295·3-s + 2.37·4-s + 0.447·5-s − 0.543·6-s + 0.135·7-s − 2.53·8-s − 0.912·9-s − 0.822·10-s + 0.301·11-s + 0.703·12-s − 1.93·13-s − 0.248·14-s + 0.132·15-s + 2.28·16-s + 0.439·17-s + 1.67·18-s + 0.229·19-s + 1.06·20-s + 0.0399·21-s − 0.554·22-s − 0.631·23-s − 0.749·24-s + 0.200·25-s + 3.55·26-s − 0.565·27-s + 0.321·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.3899389628\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3899389628\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - 25T \) |
| 11 | \( 1 - 121T \) |
| 19 | \( 1 - 361T \) |
| good | 2 | \( 1 + 10.3T + 32T^{2} \) |
| 3 | \( 1 - 4.60T + 243T^{2} \) |
| 7 | \( 1 - 17.5T + 1.68e4T^{2} \) |
| 13 | \( 1 + 1.17e3T + 3.71e5T^{2} \) |
| 17 | \( 1 - 523.T + 1.41e6T^{2} \) |
| 23 | \( 1 + 1.60e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 8.61e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 6.21e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.11e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 9.08e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 2.34e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.15e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 7.33e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.38e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.89e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 5.38e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 6.52e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.52e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 3.77e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.35e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 9.34e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 4.15e4T + 8.58e9T^{2} \) |
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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
−9.320114169524297571310808589707, −8.526335427834919467833237964691, −7.58117141362683234383147525781, −7.22704163022056126974196380431, −6.03207918501057152946005055239, −5.20150059154732683376872213721, −3.41401400368630117623587558362, −2.33090988132671223003979637199, −1.76724402952007549674965010400, −0.33301670359579470060200488665,
0.33301670359579470060200488665, 1.76724402952007549674965010400, 2.33090988132671223003979637199, 3.41401400368630117623587558362, 5.20150059154732683376872213721, 6.03207918501057152946005055239, 7.22704163022056126974196380431, 7.58117141362683234383147525781, 8.526335427834919467833237964691, 9.320114169524297571310808589707
