| L(s) = 1 | + 2·2-s + 6.85·3-s + 4·4-s + 13.7·6-s + 30.5·7-s + 8·8-s + 19.9·9-s + 20.3·11-s + 27.4·12-s + 3.87·13-s + 61.1·14-s + 16·16-s + 17·17-s + 39.8·18-s − 34.2·19-s + 209.·21-s + 40.7·22-s − 52.6·23-s + 54.8·24-s + 7.75·26-s − 48.3·27-s + 122.·28-s + 295.·29-s − 226.·31-s + 32·32-s + 139.·33-s + 34·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.31·3-s + 0.5·4-s + 0.932·6-s + 1.65·7-s + 0.353·8-s + 0.738·9-s + 0.558·11-s + 0.659·12-s + 0.0827·13-s + 1.16·14-s + 0.250·16-s + 0.242·17-s + 0.522·18-s − 0.413·19-s + 2.17·21-s + 0.394·22-s − 0.477·23-s + 0.466·24-s + 0.0585·26-s − 0.344·27-s + 0.825·28-s + 1.89·29-s − 1.31·31-s + 0.176·32-s + 0.735·33-s + 0.171·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 850 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(6.555642638\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.555642638\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 5 | \( 1 \) |
| 17 | \( 1 - 17T \) |
| good | 3 | \( 1 - 6.85T + 27T^{2} \) |
| 7 | \( 1 - 30.5T + 343T^{2} \) |
| 11 | \( 1 - 20.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 3.87T + 2.19e3T^{2} \) |
| 19 | \( 1 + 34.2T + 6.85e3T^{2} \) |
| 23 | \( 1 + 52.6T + 1.21e4T^{2} \) |
| 29 | \( 1 - 295.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 226.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 32.6T + 5.06e4T^{2} \) |
| 41 | \( 1 - 50.7T + 6.89e4T^{2} \) |
| 43 | \( 1 + 41.8T + 7.95e4T^{2} \) |
| 47 | \( 1 + 550.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 163.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 355.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 49.1T + 2.26e5T^{2} \) |
| 67 | \( 1 - 795.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 30.3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 820.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.05e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 572.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.18e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 232.T + 9.12e5T^{2} \) |
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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
−9.707057851689937103395541485571, −8.667404546650489469806827433138, −8.162751274485048616705370393824, −7.44140574857522521505593574406, −6.31539113938886245981656337158, −5.10220644683103940707349056638, −4.31262197967526786910824028434, −3.38477497926311057984733034887, −2.25690449250141518218407144494, −1.43681271850698791822868013528,
1.43681271850698791822868013528, 2.25690449250141518218407144494, 3.38477497926311057984733034887, 4.31262197967526786910824028434, 5.10220644683103940707349056638, 6.31539113938886245981656337158, 7.44140574857522521505593574406, 8.162751274485048616705370393824, 8.667404546650489469806827433138, 9.707057851689937103395541485571
