| L(s) = 1 | − 1.94·2-s + 3.44·3-s + 1.77·4-s + 0.802·5-s − 6.68·6-s − 1.20·7-s + 0.439·8-s + 8.84·9-s − 1.55·10-s − 3.00·11-s + 6.10·12-s + 2.47·13-s + 2.34·14-s + 2.76·15-s − 4.40·16-s − 17-s − 17.1·18-s + 6.97·19-s + 1.42·20-s − 4.14·21-s + 5.83·22-s − 3.49·23-s + 1.51·24-s − 4.35·25-s − 4.81·26-s + 20.1·27-s − 2.13·28-s + ⋯ |
| L(s) = 1 | − 1.37·2-s + 1.98·3-s + 0.886·4-s + 0.358·5-s − 2.72·6-s − 0.455·7-s + 0.155·8-s + 2.94·9-s − 0.492·10-s − 0.904·11-s + 1.76·12-s + 0.687·13-s + 0.625·14-s + 0.712·15-s − 1.10·16-s − 0.242·17-s − 4.04·18-s + 1.60·19-s + 0.318·20-s − 0.905·21-s + 1.24·22-s − 0.729·23-s + 0.308·24-s − 0.871·25-s − 0.944·26-s + 3.87·27-s − 0.404·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.596007624\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.596007624\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 + T \) |
| 43 | \( 1 - T \) |
| good | 2 | \( 1 + 1.94T + 2T^{2} \) |
| 3 | \( 1 - 3.44T + 3T^{2} \) |
| 5 | \( 1 - 0.802T + 5T^{2} \) |
| 7 | \( 1 + 1.20T + 7T^{2} \) |
| 11 | \( 1 + 3.00T + 11T^{2} \) |
| 13 | \( 1 - 2.47T + 13T^{2} \) |
| 19 | \( 1 - 6.97T + 19T^{2} \) |
| 23 | \( 1 + 3.49T + 23T^{2} \) |
| 29 | \( 1 - 10.3T + 29T^{2} \) |
| 31 | \( 1 - 3.97T + 31T^{2} \) |
| 37 | \( 1 + 1.02T + 37T^{2} \) |
| 41 | \( 1 + 5.47T + 41T^{2} \) |
| 47 | \( 1 + 5.14T + 47T^{2} \) |
| 53 | \( 1 - 4.20T + 53T^{2} \) |
| 59 | \( 1 + 12.2T + 59T^{2} \) |
| 61 | \( 1 - 4.73T + 61T^{2} \) |
| 67 | \( 1 + 7.94T + 67T^{2} \) |
| 71 | \( 1 + 3.68T + 71T^{2} \) |
| 73 | \( 1 - 8.84T + 73T^{2} \) |
| 79 | \( 1 - 7.36T + 79T^{2} \) |
| 83 | \( 1 + 7.11T + 83T^{2} \) |
| 89 | \( 1 - 2.45T + 89T^{2} \) |
| 97 | \( 1 + 3.37T + 97T^{2} \) |
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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
−9.912963565236992329716487198289, −9.553933480129037496884001426387, −8.609305238342809960499865292908, −8.111620220834174804951934967688, −7.46740566196553979594735039788, −6.48939906983230613783998907450, −4.70437987011970859998023880098, −3.39689505427134852719698885595, −2.49217997030486046005665222683, −1.36061377852775121075815215278,
1.36061377852775121075815215278, 2.49217997030486046005665222683, 3.39689505427134852719698885595, 4.70437987011970859998023880098, 6.48939906983230613783998907450, 7.46740566196553979594735039788, 8.111620220834174804951934967688, 8.609305238342809960499865292908, 9.553933480129037496884001426387, 9.912963565236992329716487198289
