| L(s) = 1 | − 2-s + 2.62·3-s + 4-s − 2.62·6-s − 0.864·7-s − 8-s + 3.89·9-s + 2·11-s + 2.62·12-s + 2.62·13-s + 0.864·14-s + 16-s − 17-s − 3.89·18-s − 0.896·19-s − 2.27·21-s − 2·22-s + 3.13·23-s − 2.62·24-s − 2.62·26-s + 2.35·27-s − 0.864·28-s + 9.49·29-s + 9.01·31-s − 32-s + 5.25·33-s + 34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.51·3-s + 0.5·4-s − 1.07·6-s − 0.326·7-s − 0.353·8-s + 1.29·9-s + 0.603·11-s + 0.758·12-s + 0.728·13-s + 0.231·14-s + 0.250·16-s − 0.242·17-s − 0.918·18-s − 0.205·19-s − 0.495·21-s − 0.426·22-s + 0.653·23-s − 0.536·24-s − 0.515·26-s + 0.453·27-s − 0.163·28-s + 1.76·29-s + 1.61·31-s − 0.176·32-s + 0.914·33-s + 0.171·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.962358196\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.962358196\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 - 2.62T + 3T^{2} \) |
| 7 | \( 1 + 0.864T + 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 - 2.62T + 13T^{2} \) |
| 19 | \( 1 + 0.896T + 19T^{2} \) |
| 23 | \( 1 - 3.13T + 23T^{2} \) |
| 29 | \( 1 - 9.49T + 29T^{2} \) |
| 31 | \( 1 - 9.01T + 31T^{2} \) |
| 37 | \( 1 + 10.1T + 37T^{2} \) |
| 41 | \( 1 - 9.52T + 41T^{2} \) |
| 43 | \( 1 + 7.25T + 43T^{2} \) |
| 47 | \( 1 - 10.4T + 47T^{2} \) |
| 53 | \( 1 + 11.4T + 53T^{2} \) |
| 59 | \( 1 + 4.14T + 59T^{2} \) |
| 61 | \( 1 - 3.28T + 61T^{2} \) |
| 67 | \( 1 + 1.25T + 67T^{2} \) |
| 71 | \( 1 + 12.5T + 71T^{2} \) |
| 73 | \( 1 - 2.62T + 73T^{2} \) |
| 79 | \( 1 + 7.91T + 79T^{2} \) |
| 83 | \( 1 - 4.20T + 83T^{2} \) |
| 89 | \( 1 - 4.14T + 89T^{2} \) |
| 97 | \( 1 - 6.14T + 97T^{2} \) |
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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
−9.952061447068853868953915050808, −9.107058156720923522872796858893, −8.623079858763230314532995126710, −7.968209318191468384501583390149, −6.93687332909939608527653667037, −6.22082612122602334819725178652, −4.54421004179700306902625924931, −3.41471372271775913553698491075, −2.63977837872655163018759486397, −1.33338004419201594827479386392,
1.33338004419201594827479386392, 2.63977837872655163018759486397, 3.41471372271775913553698491075, 4.54421004179700306902625924931, 6.22082612122602334819725178652, 6.93687332909939608527653667037, 7.968209318191468384501583390149, 8.623079858763230314532995126710, 9.107058156720923522872796858893, 9.952061447068853868953915050808
