| L(s) = 1 | + 2-s − 2.73·3-s + 4-s − 0.732·5-s − 2.73·6-s + 7-s + 8-s + 4.46·9-s − 0.732·10-s + 3.46·11-s − 2.73·12-s + 5.46·13-s + 14-s + 2·15-s + 16-s + 2.73·17-s + 4.46·18-s − 2·19-s − 0.732·20-s − 2.73·21-s + 3.46·22-s + 23-s − 2.73·24-s − 4.46·25-s + 5.46·26-s − 3.99·27-s + 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.57·3-s + 0.5·4-s − 0.327·5-s − 1.11·6-s + 0.377·7-s + 0.353·8-s + 1.48·9-s − 0.231·10-s + 1.04·11-s − 0.788·12-s + 1.51·13-s + 0.267·14-s + 0.516·15-s + 0.250·16-s + 0.662·17-s + 1.05·18-s − 0.458·19-s − 0.163·20-s − 0.596·21-s + 0.738·22-s + 0.208·23-s − 0.557·24-s − 0.892·25-s + 1.07·26-s − 0.769·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.327023431\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.327023431\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 + 2.73T + 3T^{2} \) |
| 5 | \( 1 + 0.732T + 5T^{2} \) |
| 11 | \( 1 - 3.46T + 11T^{2} \) |
| 13 | \( 1 - 5.46T + 13T^{2} \) |
| 17 | \( 1 - 2.73T + 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 29 | \( 1 - 8T + 29T^{2} \) |
| 31 | \( 1 + 10.1T + 31T^{2} \) |
| 37 | \( 1 - 7.46T + 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 - 2.19T + 47T^{2} \) |
| 53 | \( 1 + 3.46T + 53T^{2} \) |
| 59 | \( 1 - 5.66T + 59T^{2} \) |
| 61 | \( 1 + 2.19T + 61T^{2} \) |
| 67 | \( 1 + 11.4T + 67T^{2} \) |
| 71 | \( 1 + 12.3T + 71T^{2} \) |
| 73 | \( 1 + 0.535T + 73T^{2} \) |
| 79 | \( 1 + 12.9T + 79T^{2} \) |
| 83 | \( 1 - 14.3T + 83T^{2} \) |
| 89 | \( 1 - 3.80T + 89T^{2} \) |
| 97 | \( 1 + 5.26T + 97T^{2} \) |
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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
−11.58633516157768698972319476615, −11.11381917331582317230341312977, −10.23218977759189274663605114359, −8.773354454877926775250115319318, −7.44881522608849169141167226406, −6.31971864967619896396942253525, −5.82809701286436456124192797961, −4.62123766791621609842417052097, −3.70438461595417975502626028657, −1.30103763604643903910844266711,
1.30103763604643903910844266711, 3.70438461595417975502626028657, 4.62123766791621609842417052097, 5.82809701286436456124192797961, 6.31971864967619896396942253525, 7.44881522608849169141167226406, 8.773354454877926775250115319318, 10.23218977759189274663605114359, 11.11381917331582317230341312977, 11.58633516157768698972319476615
