| L(s) = 1 | − 1.08·2-s − 0.806·3-s − 0.828·4-s + 1.00·5-s + 0.872·6-s − 2.77·7-s + 3.06·8-s − 2.34·9-s − 1.08·10-s − 5.77·11-s + 0.667·12-s + 13-s + 3.00·14-s − 0.808·15-s − 1.65·16-s + 4.28·17-s + 2.54·18-s − 4.41·19-s − 0.830·20-s + 2.23·21-s + 6.24·22-s − 0.981·23-s − 2.46·24-s − 3.99·25-s − 1.08·26-s + 4.31·27-s + 2.29·28-s + ⋯ |
| L(s) = 1 | − 0.765·2-s − 0.465·3-s − 0.414·4-s + 0.448·5-s + 0.356·6-s − 1.04·7-s + 1.08·8-s − 0.783·9-s − 0.343·10-s − 1.74·11-s + 0.192·12-s + 0.277·13-s + 0.802·14-s − 0.208·15-s − 0.414·16-s + 1.03·17-s + 0.599·18-s − 1.01·19-s − 0.185·20-s + 0.487·21-s + 1.33·22-s − 0.204·23-s − 0.503·24-s − 0.798·25-s − 0.212·26-s + 0.830·27-s + 0.434·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3871970428\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3871970428\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 - T \) |
| 103 | \( 1 + T \) |
| good | 2 | \( 1 + 1.08T + 2T^{2} \) |
| 3 | \( 1 + 0.806T + 3T^{2} \) |
| 5 | \( 1 - 1.00T + 5T^{2} \) |
| 7 | \( 1 + 2.77T + 7T^{2} \) |
| 11 | \( 1 + 5.77T + 11T^{2} \) |
| 17 | \( 1 - 4.28T + 17T^{2} \) |
| 19 | \( 1 + 4.41T + 19T^{2} \) |
| 23 | \( 1 + 0.981T + 23T^{2} \) |
| 29 | \( 1 + 6.36T + 29T^{2} \) |
| 31 | \( 1 - 5.65T + 31T^{2} \) |
| 37 | \( 1 - 10.6T + 37T^{2} \) |
| 41 | \( 1 + 4.28T + 41T^{2} \) |
| 43 | \( 1 + 3.11T + 43T^{2} \) |
| 47 | \( 1 - 5.36T + 47T^{2} \) |
| 53 | \( 1 - 4.88T + 53T^{2} \) |
| 59 | \( 1 - 5.37T + 59T^{2} \) |
| 61 | \( 1 + 4.51T + 61T^{2} \) |
| 67 | \( 1 - 4.14T + 67T^{2} \) |
| 71 | \( 1 + 13.2T + 71T^{2} \) |
| 73 | \( 1 + 0.834T + 73T^{2} \) |
| 79 | \( 1 - 3.55T + 79T^{2} \) |
| 83 | \( 1 + 8.09T + 83T^{2} \) |
| 89 | \( 1 - 10.9T + 89T^{2} \) |
| 97 | \( 1 - 14.3T + 97T^{2} \) |
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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
−9.821743950587972167622561761041, −8.824278134304368779890188329105, −8.114056956689080821701365319102, −7.41311359638369613331869752230, −6.10019417713470953668812965656, −5.67147729699942974119757429693, −4.66860060164864356212062574507, −3.37553205876412065026855550933, −2.27529172896107045859874415355, −0.49579260929895019975015584638,
0.49579260929895019975015584638, 2.27529172896107045859874415355, 3.37553205876412065026855550933, 4.66860060164864356212062574507, 5.67147729699942974119757429693, 6.10019417713470953668812965656, 7.41311359638369613331869752230, 8.114056956689080821701365319102, 8.824278134304368779890188329105, 9.821743950587972167622561761041
