| L(s) = 1 | + 2-s − 3-s + 4-s + 2.09·5-s − 6-s + 7-s + 8-s + 9-s + 2.09·10-s − 1.01·11-s − 12-s + 1.37·13-s + 14-s − 2.09·15-s + 16-s + 6.65·17-s + 18-s − 1.25·19-s + 2.09·20-s − 21-s − 1.01·22-s + 5.57·23-s − 24-s − 0.622·25-s + 1.37·26-s − 27-s + 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.935·5-s − 0.408·6-s + 0.377·7-s + 0.353·8-s + 0.333·9-s + 0.661·10-s − 0.305·11-s − 0.288·12-s + 0.380·13-s + 0.267·14-s − 0.540·15-s + 0.250·16-s + 1.61·17-s + 0.235·18-s − 0.287·19-s + 0.467·20-s − 0.218·21-s − 0.216·22-s + 1.16·23-s − 0.204·24-s − 0.124·25-s + 0.268·26-s − 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.938241246\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.938241246\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 191 | \( 1 - T \) |
| good | 5 | \( 1 - 2.09T + 5T^{2} \) |
| 11 | \( 1 + 1.01T + 11T^{2} \) |
| 13 | \( 1 - 1.37T + 13T^{2} \) |
| 17 | \( 1 - 6.65T + 17T^{2} \) |
| 19 | \( 1 + 1.25T + 19T^{2} \) |
| 23 | \( 1 - 5.57T + 23T^{2} \) |
| 29 | \( 1 - 4.22T + 29T^{2} \) |
| 31 | \( 1 - 4.64T + 31T^{2} \) |
| 37 | \( 1 + 3.39T + 37T^{2} \) |
| 41 | \( 1 + 0.205T + 41T^{2} \) |
| 43 | \( 1 - 0.130T + 43T^{2} \) |
| 47 | \( 1 + 4.38T + 47T^{2} \) |
| 53 | \( 1 + 7.54T + 53T^{2} \) |
| 59 | \( 1 + 11.5T + 59T^{2} \) |
| 61 | \( 1 - 2.15T + 61T^{2} \) |
| 67 | \( 1 - 13.2T + 67T^{2} \) |
| 71 | \( 1 - 8.44T + 71T^{2} \) |
| 73 | \( 1 - 3.50T + 73T^{2} \) |
| 79 | \( 1 - 15.2T + 79T^{2} \) |
| 83 | \( 1 - 4.94T + 83T^{2} \) |
| 89 | \( 1 + 10.6T + 89T^{2} \) |
| 97 | \( 1 + 17.7T + 97T^{2} \) |
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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
−7.84553702386678769877198712464, −6.77850618224296170015709242878, −6.41404088105003433134377506063, −5.52342047789308083767097355478, −5.24840253995818510209038701084, −4.50350303625433983682656931387, −3.50321319310704097742204321829, −2.76210306223200386120348139942, −1.74820936834990252256178942070, −0.976711580531831621534106850786,
0.976711580531831621534106850786, 1.74820936834990252256178942070, 2.76210306223200386120348139942, 3.50321319310704097742204321829, 4.50350303625433983682656931387, 5.24840253995818510209038701084, 5.52342047789308083767097355478, 6.41404088105003433134377506063, 6.77850618224296170015709242878, 7.84553702386678769877198712464
