| L(s) = 1 | − 2.50·2-s + 2.99·3-s + 4.25·4-s + 4.07·5-s − 7.50·6-s − 1.09·7-s − 5.63·8-s + 5.99·9-s − 10.1·10-s + 3.69·11-s + 12.7·12-s + 13-s + 2.73·14-s + 12.2·15-s + 5.59·16-s − 6.18·17-s − 14.9·18-s − 0.160·19-s + 17.3·20-s − 3.27·21-s − 9.24·22-s − 4.76·23-s − 16.9·24-s + 11.5·25-s − 2.50·26-s + 8.98·27-s − 4.65·28-s + ⋯ |
| L(s) = 1 | − 1.76·2-s + 1.73·3-s + 2.12·4-s + 1.82·5-s − 3.06·6-s − 0.413·7-s − 1.99·8-s + 1.99·9-s − 3.22·10-s + 1.11·11-s + 3.68·12-s + 0.277·13-s + 0.730·14-s + 3.15·15-s + 1.39·16-s − 1.50·17-s − 3.53·18-s − 0.0369·19-s + 3.87·20-s − 0.715·21-s − 1.97·22-s − 0.993·23-s − 3.45·24-s + 2.31·25-s − 0.490·26-s + 1.72·27-s − 0.879·28-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
| \(L(1)\) |
\(\approx\) |
\(1.924437055\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.924437055\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.50T + 2T^{2} \) |
| 3 | \( 1 - 2.99T + 3T^{2} \) |
| 5 | \( 1 - 4.07T + 5T^{2} \) |
| 7 | \( 1 + 1.09T + 7T^{2} \) |
| 11 | \( 1 - 3.69T + 11T^{2} \) |
| 17 | \( 1 + 6.18T + 17T^{2} \) |
| 19 | \( 1 + 0.160T + 19T^{2} \) |
| 23 | \( 1 + 4.76T + 23T^{2} \) |
| 29 | \( 1 + 5.14T + 29T^{2} \) |
| 31 | \( 1 - 6.88T + 31T^{2} \) |
| 37 | \( 1 + 5.59T + 37T^{2} \) |
| 41 | \( 1 - 11.0T + 41T^{2} \) |
| 43 | \( 1 + 4.47T + 43T^{2} \) |
| 47 | \( 1 - 8.77T + 47T^{2} \) |
| 53 | \( 1 - 0.346T + 53T^{2} \) |
| 59 | \( 1 - 6.68T + 59T^{2} \) |
| 61 | \( 1 - 8.34T + 61T^{2} \) |
| 67 | \( 1 + 9.72T + 67T^{2} \) |
| 71 | \( 1 + 9.43T + 71T^{2} \) |
| 73 | \( 1 + 13.7T + 73T^{2} \) |
| 79 | \( 1 + 2.33T + 79T^{2} \) |
| 83 | \( 1 - 1.69T + 83T^{2} \) |
| 89 | \( 1 + 10.1T + 89T^{2} \) |
| 97 | \( 1 + 1.94T + 97T^{2} \) |
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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
−9.404744476251413857249568787454, −8.894701124402344678720025728953, −8.553865467172281376605250451771, −7.39201067974581994489133285379, −6.65257500425412454915687527527, −6.03524335959741346989089328600, −4.16012850253478325633505858011, −2.78956831339698522742353110578, −2.10732095335496335865405515927, −1.42239117305986417779581528240,
1.42239117305986417779581528240, 2.10732095335496335865405515927, 2.78956831339698522742353110578, 4.16012850253478325633505858011, 6.03524335959741346989089328600, 6.65257500425412454915687527527, 7.39201067974581994489133285379, 8.553865467172281376605250451771, 8.894701124402344678720025728953, 9.404744476251413857249568787454
