| L(s) = 1 | − 4.58·3-s + 1.52i·7-s + 12.0·9-s + 19.7·11-s + 16.2·13-s − 15.9i·17-s + (18.9 + 1.78i)19-s − 7.01i·21-s − 37.5i·23-s − 13.9·27-s − 31.5i·29-s + 38.9i·31-s − 90.6·33-s − 39.1·37-s − 74.4·39-s + ⋯ |
| L(s) = 1 | − 1.52·3-s + 0.218i·7-s + 1.33·9-s + 1.79·11-s + 1.24·13-s − 0.939i·17-s + (0.995 + 0.0937i)19-s − 0.334i·21-s − 1.63i·23-s − 0.517·27-s − 1.08i·29-s + 1.25i·31-s − 2.74·33-s − 1.05·37-s − 1.90·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.848 + 0.529i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.848 + 0.529i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.436190235\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.436190235\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + (-18.9 - 1.78i)T \) |
| good | 3 | \( 1 + 4.58T + 9T^{2} \) |
| 7 | \( 1 - 1.52iT - 49T^{2} \) |
| 11 | \( 1 - 19.7T + 121T^{2} \) |
| 13 | \( 1 - 16.2T + 169T^{2} \) |
| 17 | \( 1 + 15.9iT - 289T^{2} \) |
| 23 | \( 1 + 37.5iT - 529T^{2} \) |
| 29 | \( 1 + 31.5iT - 841T^{2} \) |
| 31 | \( 1 - 38.9iT - 961T^{2} \) |
| 37 | \( 1 + 39.1T + 1.36e3T^{2} \) |
| 41 | \( 1 - 69.4iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 16.5iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 52.0iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 16.8T + 2.80e3T^{2} \) |
| 59 | \( 1 + 92.5iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 51.5T + 3.72e3T^{2} \) |
| 67 | \( 1 - 48.7T + 4.48e3T^{2} \) |
| 71 | \( 1 + 5.51iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 62.8iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 146. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 157. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 87.6iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 115.T + 9.40e3T^{2} \) |
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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
−9.065475747919871409183663104959, −8.241675692103228421478114143629, −6.97353686358112281160420613219, −6.46714508282905057679073191721, −5.92739811310693677787022273754, −4.94886576974239945564630151367, −4.24075083605718773967299633747, −3.15576246666483389127880786562, −1.45283882066104560488392806505, −0.67171178254670716032099043054,
0.926196458077152070534482514535, 1.58251958903224512946929032039, 3.71842170650410439355984191899, 3.97203274966857416472699261835, 5.50474897083187943771539944589, 5.66745161071827652133378800138, 6.76277471411092410755463033726, 7.08288373547414459381814494469, 8.402611884381414718793757982963, 9.194097081358512510333235434282
