| L(s) = 1 | − 0.172·2-s − 1.97·4-s + 3.73·5-s + 3.03·7-s + 0.686·8-s − 0.646·10-s + 2.49·11-s − 0.765·13-s − 0.524·14-s + 3.82·16-s − 4.62·17-s − 0.611·19-s − 7.36·20-s − 0.431·22-s − 6.52·23-s + 8.96·25-s + 0.132·26-s − 5.97·28-s + 6.55·29-s + 6.55·31-s − 2.03·32-s + 0.799·34-s + 11.3·35-s + 4.95·37-s + 0.105·38-s + 2.56·40-s − 5.26·41-s + ⋯ |
| L(s) = 1 | − 0.122·2-s − 0.985·4-s + 1.67·5-s + 1.14·7-s + 0.242·8-s − 0.204·10-s + 0.751·11-s − 0.212·13-s − 0.140·14-s + 0.955·16-s − 1.12·17-s − 0.140·19-s − 1.64·20-s − 0.0918·22-s − 1.36·23-s + 1.79·25-s + 0.0259·26-s − 1.12·28-s + 1.21·29-s + 1.17·31-s − 0.359·32-s + 0.137·34-s + 1.91·35-s + 0.815·37-s + 0.0171·38-s + 0.405·40-s − 0.821·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.758063321\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.758063321\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + 0.172T + 2T^{2} \) |
| 5 | \( 1 - 3.73T + 5T^{2} \) |
| 7 | \( 1 - 3.03T + 7T^{2} \) |
| 11 | \( 1 - 2.49T + 11T^{2} \) |
| 13 | \( 1 + 0.765T + 13T^{2} \) |
| 17 | \( 1 + 4.62T + 17T^{2} \) |
| 19 | \( 1 + 0.611T + 19T^{2} \) |
| 23 | \( 1 + 6.52T + 23T^{2} \) |
| 29 | \( 1 - 6.55T + 29T^{2} \) |
| 31 | \( 1 - 6.55T + 31T^{2} \) |
| 37 | \( 1 - 4.95T + 37T^{2} \) |
| 41 | \( 1 + 5.26T + 41T^{2} \) |
| 43 | \( 1 - 5.57T + 43T^{2} \) |
| 47 | \( 1 + 1.10T + 47T^{2} \) |
| 53 | \( 1 + 8.84T + 53T^{2} \) |
| 59 | \( 1 - 11.8T + 59T^{2} \) |
| 61 | \( 1 - 8.18T + 61T^{2} \) |
| 67 | \( 1 + 1.21T + 67T^{2} \) |
| 71 | \( 1 - 4.91T + 71T^{2} \) |
| 73 | \( 1 - 4.29T + 73T^{2} \) |
| 79 | \( 1 + 11.7T + 79T^{2} \) |
| 83 | \( 1 + 9.01T + 83T^{2} \) |
| 89 | \( 1 + 7.53T + 89T^{2} \) |
| 97 | \( 1 + 0.948T + 97T^{2} \) |
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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
−10.06679594388672865004332037970, −9.663321466006005378871013719024, −8.684545512313086438583177907662, −8.176704561463835200651532162476, −6.70308118585673731294138373335, −5.87848157104309260851657030202, −4.91459436124261812729335569606, −4.22799385965916839065711731577, −2.39085727163418138182106927865, −1.31823066914459091943234665122,
1.31823066914459091943234665122, 2.39085727163418138182106927865, 4.22799385965916839065711731577, 4.91459436124261812729335569606, 5.87848157104309260851657030202, 6.70308118585673731294138373335, 8.176704561463835200651532162476, 8.684545512313086438583177907662, 9.663321466006005378871013719024, 10.06679594388672865004332037970
