| L(s) = 1 | − 241.·2-s + 570.·3-s + 4.21e4·4-s − 1.14e5i·5-s − 1.38e5·6-s − 2.70e5i·7-s − 6.23e6·8-s − 4.45e6·9-s + 2.76e7i·10-s − 3.28e7i·11-s + 2.40e7·12-s − 1.99e7·13-s + 6.55e7i·14-s − 6.51e7i·15-s + 8.18e8·16-s − 6.85e8i·17-s + ⋯ |
| L(s) = 1 | − 1.89·2-s + 0.261·3-s + 2.57·4-s − 1.46i·5-s − 0.493·6-s − 0.328i·7-s − 2.97·8-s − 0.931·9-s + 2.76i·10-s − 1.68i·11-s + 0.671·12-s − 0.317·13-s + 0.621i·14-s − 0.381i·15-s + 3.04·16-s − 1.67i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.863 - 0.504i)\, \overline{\Lambda}(15-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+7) \, L(s)\cr =\mathstrut & (-0.863 - 0.504i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{15}{2})\) |
\(\approx\) |
\(0.4594200009\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4594200009\) |
| \(L(8)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 + (-2.93e9 - 1.71e9i)T \) |
| good | 2 | \( 1 + 241.T + 1.63e4T^{2} \) |
| 3 | \( 1 - 570.T + 4.78e6T^{2} \) |
| 5 | \( 1 + 1.14e5iT - 6.10e9T^{2} \) |
| 7 | \( 1 + 2.70e5iT - 6.78e11T^{2} \) |
| 11 | \( 1 + 3.28e7iT - 3.79e14T^{2} \) |
| 13 | \( 1 + 1.99e7T + 3.93e15T^{2} \) |
| 17 | \( 1 + 6.85e8iT - 1.68e17T^{2} \) |
| 19 | \( 1 + 9.52e6iT - 7.99e17T^{2} \) |
| 29 | \( 1 - 1.79e10T + 2.97e20T^{2} \) |
| 31 | \( 1 + 2.19e10T + 7.56e20T^{2} \) |
| 37 | \( 1 + 1.33e11iT - 9.01e21T^{2} \) |
| 41 | \( 1 + 9.37e10T + 3.79e22T^{2} \) |
| 43 | \( 1 + 1.93e10iT - 7.38e22T^{2} \) |
| 47 | \( 1 - 6.89e11T + 2.56e23T^{2} \) |
| 53 | \( 1 - 1.73e12iT - 1.37e24T^{2} \) |
| 59 | \( 1 + 3.24e12T + 6.19e24T^{2} \) |
| 61 | \( 1 + 3.95e11iT - 9.87e24T^{2} \) |
| 67 | \( 1 - 1.16e12iT - 3.67e25T^{2} \) |
| 71 | \( 1 + 1.13e13T + 8.27e25T^{2} \) |
| 73 | \( 1 + 4.90e12T + 1.22e26T^{2} \) |
| 79 | \( 1 - 1.94e13iT - 3.68e26T^{2} \) |
| 83 | \( 1 + 9.99e11iT - 7.36e26T^{2} \) |
| 89 | \( 1 + 2.17e13iT - 1.95e27T^{2} \) |
| 97 | \( 1 + 5.75e13iT - 6.52e27T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.83520786508689808748502237084, −11.95870976705034522350561055780, −10.91735254269280115398488585590, −9.148161160614713423672339741226, −8.779095636220035517542573044216, −7.52278252099579507985171661132, −5.61453018105043719206235149246, −2.83361992627694733259224003508, −0.984178539866808538367386882813, −0.31674130463075066849535826768,
1.90060078725075390578843087104, 2.88615690238108877599623522171, 6.37870691799831548326068110086, 7.38594057109422060544340680902, 8.645632528601558838879368043197, 10.01985706086910420528799855232, 10.80301969045864600787921223766, 12.11278988388875963628353317595, 14.81980130960855233714329223728, 15.18042603299554507406581379193
