| L(s) = 1 | − 0.800·2-s + 0.744·3-s − 1.35·4-s − 1.40·5-s − 0.596·6-s − 4.75·7-s + 2.68·8-s − 2.44·9-s + 1.12·10-s − 2.66·11-s − 1.01·12-s − 0.617·13-s + 3.80·14-s − 1.04·15-s + 0.565·16-s − 5.91·17-s + 1.95·18-s + 1.79·19-s + 1.90·20-s − 3.54·21-s + 2.13·22-s + 2.38·23-s + 2.00·24-s − 3.03·25-s + 0.494·26-s − 4.05·27-s + 6.46·28-s + ⋯ |
| L(s) = 1 | − 0.566·2-s + 0.429·3-s − 0.679·4-s − 0.626·5-s − 0.243·6-s − 1.79·7-s + 0.950·8-s − 0.815·9-s + 0.354·10-s − 0.802·11-s − 0.292·12-s − 0.171·13-s + 1.01·14-s − 0.269·15-s + 0.141·16-s − 1.43·17-s + 0.461·18-s + 0.411·19-s + 0.425·20-s − 0.773·21-s + 0.454·22-s + 0.497·23-s + 0.408·24-s − 0.607·25-s + 0.0970·26-s − 0.780·27-s + 1.22·28-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
| \(L(1)\) |
\(\approx\) |
\(0.2465433281\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2465433281\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.800T + 2T^{2} \) |
| 3 | \( 1 - 0.744T + 3T^{2} \) |
| 5 | \( 1 + 1.40T + 5T^{2} \) |
| 7 | \( 1 + 4.75T + 7T^{2} \) |
| 11 | \( 1 + 2.66T + 11T^{2} \) |
| 13 | \( 1 + 0.617T + 13T^{2} \) |
| 17 | \( 1 + 5.91T + 17T^{2} \) |
| 19 | \( 1 - 1.79T + 19T^{2} \) |
| 23 | \( 1 - 2.38T + 23T^{2} \) |
| 29 | \( 1 - 1.88T + 29T^{2} \) |
| 31 | \( 1 + 7.49T + 31T^{2} \) |
| 37 | \( 1 - 2.22T + 37T^{2} \) |
| 41 | \( 1 - 2.62T + 41T^{2} \) |
| 47 | \( 1 - 8.96T + 47T^{2} \) |
| 53 | \( 1 + 13.7T + 53T^{2} \) |
| 59 | \( 1 + 6.78T + 59T^{2} \) |
| 61 | \( 1 - 4.13T + 61T^{2} \) |
| 67 | \( 1 - 5.63T + 67T^{2} \) |
| 71 | \( 1 - 8.58T + 71T^{2} \) |
| 73 | \( 1 - 8.51T + 73T^{2} \) |
| 79 | \( 1 + 6.19T + 79T^{2} \) |
| 83 | \( 1 + 1.27T + 83T^{2} \) |
| 89 | \( 1 - 9.46T + 89T^{2} \) |
| 97 | \( 1 + 4.50T + 97T^{2} \) |
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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
−9.329055975086804120352215887482, −8.590449543584646494356912435324, −7.82028107463079993467330023857, −7.11017225753880528754553633712, −6.12867316443263541746862200378, −5.18654648420847631850542195293, −4.07066249457081808114355653497, −3.32464575261546318329634371200, −2.41413698330330238976072172138, −0.33501389734780362823164372758,
0.33501389734780362823164372758, 2.41413698330330238976072172138, 3.32464575261546318329634371200, 4.07066249457081808114355653497, 5.18654648420847631850542195293, 6.12867316443263541746862200378, 7.11017225753880528754553633712, 7.82028107463079993467330023857, 8.590449543584646494356912435324, 9.329055975086804120352215887482
