L(s) = 1 | + 2.61·2-s + 0.381·3-s + 4.85·4-s + 3.23·5-s + 6-s − 0.236·7-s + 7.47·8-s − 2.85·9-s + 8.47·10-s − 1.38·11-s + 1.85·12-s + 3.61·13-s − 0.618·14-s + 1.23·15-s + 9.85·16-s − 5.09·17-s − 7.47·18-s − 3.23·19-s + 15.7·20-s − 0.0901·21-s − 3.61·22-s + 6.61·23-s + 2.85·24-s + 5.47·25-s + 9.47·26-s − 2.23·27-s − 1.14·28-s + ⋯ |
L(s) = 1 | + 1.85·2-s + 0.220·3-s + 2.42·4-s + 1.44·5-s + 0.408·6-s − 0.0892·7-s + 2.64·8-s − 0.951·9-s + 2.67·10-s − 0.416·11-s + 0.535·12-s + 1.00·13-s − 0.165·14-s + 0.319·15-s + 2.46·16-s − 1.23·17-s − 1.76·18-s − 0.742·19-s + 3.51·20-s − 0.0196·21-s − 0.771·22-s + 1.37·23-s + 0.582·24-s + 1.09·25-s + 1.85·26-s − 0.430·27-s − 0.216·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(6.941439053\) |
\(L(\frac12)\) |
\(\approx\) |
\(6.941439053\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 43 | \( 1 \) |
good | 2 | \( 1 - 2.61T + 2T^{2} \) |
| 3 | \( 1 - 0.381T + 3T^{2} \) |
| 5 | \( 1 - 3.23T + 5T^{2} \) |
| 7 | \( 1 + 0.236T + 7T^{2} \) |
| 11 | \( 1 + 1.38T + 11T^{2} \) |
| 13 | \( 1 - 3.61T + 13T^{2} \) |
| 17 | \( 1 + 5.09T + 17T^{2} \) |
| 19 | \( 1 + 3.23T + 19T^{2} \) |
| 23 | \( 1 - 6.61T + 23T^{2} \) |
| 29 | \( 1 + 3T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 1.85T + 37T^{2} \) |
| 41 | \( 1 - 0.527T + 41T^{2} \) |
| 47 | \( 1 - 7.85T + 47T^{2} \) |
| 53 | \( 1 + 3.61T + 53T^{2} \) |
| 59 | \( 1 + 6.09T + 59T^{2} \) |
| 61 | \( 1 - 3.85T + 61T^{2} \) |
| 67 | \( 1 - 2.85T + 67T^{2} \) |
| 71 | \( 1 + 13.7T + 71T^{2} \) |
| 73 | \( 1 - 4.85T + 73T^{2} \) |
| 79 | \( 1 + 3.61T + 79T^{2} \) |
| 83 | \( 1 + 13.0T + 83T^{2} \) |
| 89 | \( 1 + 4.85T + 89T^{2} \) |
| 97 | \( 1 + 4.76T + 97T^{2} \) |
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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
−9.157245378146195015630120050202, −8.541359632058773430479711239536, −7.21980409670916809098736502934, −6.37460183360299329862601038479, −5.93813133842221850413759612647, −5.25615959319269638699250998013, −4.41697556712553738165567929749, −3.28920948309274159165405594782, −2.55895183636128467780974768368, −1.77093037278822331842001548965,
1.77093037278822331842001548965, 2.55895183636128467780974768368, 3.28920948309274159165405594782, 4.41697556712553738165567929749, 5.25615959319269638699250998013, 5.93813133842221850413759612647, 6.37460183360299329862601038479, 7.21980409670916809098736502934, 8.541359632058773430479711239536, 9.157245378146195015630120050202