| L(s) = 1 | + 2-s − 2.58·3-s + 4-s − 3.38·5-s − 2.58·6-s − 3.95·7-s + 8-s + 3.68·9-s − 3.38·10-s + 4.46·11-s − 2.58·12-s + 1.75·13-s − 3.95·14-s + 8.74·15-s + 16-s − 1.83·17-s + 3.68·18-s + 2.92·19-s − 3.38·20-s + 10.2·21-s + 4.46·22-s + 6.29·23-s − 2.58·24-s + 6.44·25-s + 1.75·26-s − 1.76·27-s − 3.95·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.49·3-s + 0.5·4-s − 1.51·5-s − 1.05·6-s − 1.49·7-s + 0.353·8-s + 1.22·9-s − 1.06·10-s + 1.34·11-s − 0.746·12-s + 0.487·13-s − 1.05·14-s + 2.25·15-s + 0.250·16-s − 0.445·17-s + 0.867·18-s + 0.670·19-s − 0.756·20-s + 2.22·21-s + 0.952·22-s + 1.31·23-s − 0.527·24-s + 1.28·25-s + 0.344·26-s − 0.339·27-s − 0.746·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2738 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2738 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 37 | \( 1 \) |
| good | 3 | \( 1 + 2.58T + 3T^{2} \) |
| 5 | \( 1 + 3.38T + 5T^{2} \) |
| 7 | \( 1 + 3.95T + 7T^{2} \) |
| 11 | \( 1 - 4.46T + 11T^{2} \) |
| 13 | \( 1 - 1.75T + 13T^{2} \) |
| 17 | \( 1 + 1.83T + 17T^{2} \) |
| 19 | \( 1 - 2.92T + 19T^{2} \) |
| 23 | \( 1 - 6.29T + 23T^{2} \) |
| 29 | \( 1 - 1.45T + 29T^{2} \) |
| 31 | \( 1 + 2.62T + 31T^{2} \) |
| 41 | \( 1 + 1.87T + 41T^{2} \) |
| 43 | \( 1 + 8.40T + 43T^{2} \) |
| 47 | \( 1 + 11.2T + 47T^{2} \) |
| 53 | \( 1 + 3.82T + 53T^{2} \) |
| 59 | \( 1 + 0.662T + 59T^{2} \) |
| 61 | \( 1 - 6.41T + 61T^{2} \) |
| 67 | \( 1 + 6.08T + 67T^{2} \) |
| 71 | \( 1 - 3.22T + 71T^{2} \) |
| 73 | \( 1 - 9.82T + 73T^{2} \) |
| 79 | \( 1 - 11.0T + 79T^{2} \) |
| 83 | \( 1 + 5.86T + 83T^{2} \) |
| 89 | \( 1 + 18.5T + 89T^{2} \) |
| 97 | \( 1 + 8.08T + 97T^{2} \) |
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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
−8.366594190390412292424468331236, −7.17393430051974009448491990828, −6.70211932770185921745397963802, −6.31635547826700409317274745607, −5.28399633033523702517146934487, −4.51849481974698583264396801688, −3.66082704996987196392617369853, −3.20388906160444118171769792037, −1.14027906043359257070415719984, 0,
1.14027906043359257070415719984, 3.20388906160444118171769792037, 3.66082704996987196392617369853, 4.51849481974698583264396801688, 5.28399633033523702517146934487, 6.31635547826700409317274745607, 6.70211932770185921745397963802, 7.17393430051974009448491990828, 8.366594190390412292424468331236
