| L(s) = 1 | + 0.737·2-s + 2.97·3-s − 1.45·4-s + 5-s + 2.19·6-s + 1.93·7-s − 2.54·8-s + 5.84·9-s + 0.737·10-s − 2.66·11-s − 4.32·12-s − 2.07·13-s + 1.42·14-s + 2.97·15-s + 1.03·16-s + 7.48·17-s + 4.30·18-s − 2.49·19-s − 1.45·20-s + 5.74·21-s − 1.96·22-s + 7.64·23-s − 7.58·24-s + 25-s − 1.52·26-s + 8.45·27-s − 2.81·28-s + ⋯ |
| L(s) = 1 | + 0.521·2-s + 1.71·3-s − 0.727·4-s + 0.447·5-s + 0.895·6-s + 0.729·7-s − 0.901·8-s + 1.94·9-s + 0.233·10-s − 0.802·11-s − 1.24·12-s − 0.575·13-s + 0.380·14-s + 0.767·15-s + 0.257·16-s + 1.81·17-s + 1.01·18-s − 0.573·19-s − 0.325·20-s + 1.25·21-s − 0.418·22-s + 1.59·23-s − 1.54·24-s + 0.200·25-s − 0.300·26-s + 1.62·27-s − 0.531·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.735719604\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.735719604\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 31 | \( 1 \) |
| good | 2 | \( 1 - 0.737T + 2T^{2} \) |
| 3 | \( 1 - 2.97T + 3T^{2} \) |
| 7 | \( 1 - 1.93T + 7T^{2} \) |
| 11 | \( 1 + 2.66T + 11T^{2} \) |
| 13 | \( 1 + 2.07T + 13T^{2} \) |
| 17 | \( 1 - 7.48T + 17T^{2} \) |
| 19 | \( 1 + 2.49T + 19T^{2} \) |
| 23 | \( 1 - 7.64T + 23T^{2} \) |
| 29 | \( 1 + 3.59T + 29T^{2} \) |
| 37 | \( 1 + 2.84T + 37T^{2} \) |
| 41 | \( 1 - 9.86T + 41T^{2} \) |
| 43 | \( 1 - 1.32T + 43T^{2} \) |
| 47 | \( 1 + 1.50T + 47T^{2} \) |
| 53 | \( 1 - 9.18T + 53T^{2} \) |
| 59 | \( 1 - 12.3T + 59T^{2} \) |
| 61 | \( 1 - 2.67T + 61T^{2} \) |
| 67 | \( 1 + 0.787T + 67T^{2} \) |
| 71 | \( 1 + 5.23T + 71T^{2} \) |
| 73 | \( 1 + 12.4T + 73T^{2} \) |
| 79 | \( 1 + 10.5T + 79T^{2} \) |
| 83 | \( 1 - 8.30T + 83T^{2} \) |
| 89 | \( 1 - 6.26T + 89T^{2} \) |
| 97 | \( 1 - 10.1T + 97T^{2} \) |
| 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
−8.433860657723231893392582126626, −7.62292483146976901265889889678, −7.22728580724597919154212077221, −5.81500537580298051795654953383, −5.18064586656044320927353665485, −4.51034333335736371037543913532, −3.62276092161798429277131379976, −2.95638102978990703490022486352, −2.25981729075605843069802413780, −1.09737330259781234847808189479,
1.09737330259781234847808189479, 2.25981729075605843069802413780, 2.95638102978990703490022486352, 3.62276092161798429277131379976, 4.51034333335736371037543913532, 5.18064586656044320927353665485, 5.81500537580298051795654953383, 7.22728580724597919154212077221, 7.62292483146976901265889889678, 8.433860657723231893392582126626
