| L(s) = 1 | − 0.688·2-s + 1.21·3-s − 1.52·4-s − 2.31·5-s − 0.836·6-s − 4.42·7-s + 2.42·8-s − 1.52·9-s + 1.59·10-s − 11-s − 1.85·12-s − 0.0666·13-s + 3.05·14-s − 2.80·15-s + 1.37·16-s − 17-s + 1.05·18-s + 6.49·19-s + 3.52·20-s − 5.37·21-s + 0.688·22-s − 2.78·23-s + 2.94·24-s + 0.341·25-s + 0.0459·26-s − 5.49·27-s + 6.75·28-s + ⋯ |
| L(s) = 1 | − 0.487·2-s + 0.701·3-s − 0.762·4-s − 1.03·5-s − 0.341·6-s − 1.67·7-s + 0.858·8-s − 0.508·9-s + 0.503·10-s − 0.301·11-s − 0.534·12-s − 0.0184·13-s + 0.815·14-s − 0.724·15-s + 0.344·16-s − 0.242·17-s + 0.247·18-s + 1.49·19-s + 0.788·20-s − 1.17·21-s + 0.146·22-s − 0.580·23-s + 0.601·24-s + 0.0682·25-s + 0.00900·26-s − 1.05·27-s + 1.27·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 2 | \( 1 + 0.688T + 2T^{2} \) |
| 3 | \( 1 - 1.21T + 3T^{2} \) |
| 5 | \( 1 + 2.31T + 5T^{2} \) |
| 7 | \( 1 + 4.42T + 7T^{2} \) |
| 13 | \( 1 + 0.0666T + 13T^{2} \) |
| 19 | \( 1 - 6.49T + 19T^{2} \) |
| 23 | \( 1 + 2.78T + 23T^{2} \) |
| 29 | \( 1 + 2.40T + 29T^{2} \) |
| 31 | \( 1 + 7.49T + 31T^{2} \) |
| 37 | \( 1 - 7.44T + 37T^{2} \) |
| 41 | \( 1 + 4.42T + 41T^{2} \) |
| 43 | \( 1 - 11.6T + 43T^{2} \) |
| 47 | \( 1 + 1.33T + 47T^{2} \) |
| 53 | \( 1 + 12.2T + 53T^{2} \) |
| 59 | \( 1 - 1.76T + 59T^{2} \) |
| 61 | \( 1 - 2.42T + 61T^{2} \) |
| 67 | \( 1 + 13.2T + 67T^{2} \) |
| 71 | \( 1 + 1.06T + 71T^{2} \) |
| 73 | \( 1 + 1.78T + 73T^{2} \) |
| 79 | \( 1 - 6.83T + 79T^{2} \) |
| 83 | \( 1 - 2.75T + 83T^{2} \) |
| 89 | \( 1 + 8.47T + 89T^{2} \) |
| 97 | \( 1 + 7.83T + 97T^{2} \) |
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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
−12.23164647248925822319487535943, −10.99638603972622186872275007652, −9.639569588916251443498970092767, −9.250579686973059063882299130596, −8.072190170382019998248797390582, −7.35147524861074155465791736644, −5.71002900330469434750467994879, −3.98551996465400364596233619746, −3.10402286358157370951541892153, 0,
3.10402286358157370951541892153, 3.98551996465400364596233619746, 5.71002900330469434750467994879, 7.35147524861074155465791736644, 8.072190170382019998248797390582, 9.250579686973059063882299130596, 9.639569588916251443498970092767, 10.99638603972622186872275007652, 12.23164647248925822319487535943
