| L(s) = 1 | + 1.48·2-s + 1.08·3-s + 0.202·4-s − 3.88·5-s + 1.61·6-s + 2.44·7-s − 2.66·8-s − 1.81·9-s − 5.76·10-s + 11-s + 0.220·12-s − 1.10·13-s + 3.62·14-s − 4.23·15-s − 4.36·16-s + 17-s − 2.68·18-s − 2.46·19-s − 0.785·20-s + 2.66·21-s + 1.48·22-s − 5.21·23-s − 2.90·24-s + 10.1·25-s − 1.64·26-s − 5.24·27-s + 0.493·28-s + ⋯ |
| L(s) = 1 | + 1.04·2-s + 0.629·3-s + 0.101·4-s − 1.73·5-s + 0.660·6-s + 0.922·7-s − 0.943·8-s − 0.604·9-s − 1.82·10-s + 0.301·11-s + 0.0636·12-s − 0.307·13-s + 0.968·14-s − 1.09·15-s − 1.09·16-s + 0.242·17-s − 0.633·18-s − 0.565·19-s − 0.175·20-s + 0.580·21-s + 0.316·22-s − 1.08·23-s − 0.593·24-s + 2.02·25-s − 0.322·26-s − 1.00·27-s + 0.0932·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.046438001\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.046438001\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 43 | \( 1 - T \) |
| good | 2 | \( 1 - 1.48T + 2T^{2} \) |
| 3 | \( 1 - 1.08T + 3T^{2} \) |
| 5 | \( 1 + 3.88T + 5T^{2} \) |
| 7 | \( 1 - 2.44T + 7T^{2} \) |
| 13 | \( 1 + 1.10T + 13T^{2} \) |
| 19 | \( 1 + 2.46T + 19T^{2} \) |
| 23 | \( 1 + 5.21T + 23T^{2} \) |
| 29 | \( 1 - 4.22T + 29T^{2} \) |
| 31 | \( 1 + 6.14T + 31T^{2} \) |
| 37 | \( 1 - 5.89T + 37T^{2} \) |
| 41 | \( 1 - 3.75T + 41T^{2} \) |
| 47 | \( 1 - 6.32T + 47T^{2} \) |
| 53 | \( 1 - 10.9T + 53T^{2} \) |
| 59 | \( 1 + 5.69T + 59T^{2} \) |
| 61 | \( 1 + 11.0T + 61T^{2} \) |
| 67 | \( 1 - 13.2T + 67T^{2} \) |
| 71 | \( 1 - 11.2T + 71T^{2} \) |
| 73 | \( 1 + 13.3T + 73T^{2} \) |
| 79 | \( 1 - 11.9T + 79T^{2} \) |
| 83 | \( 1 + 10.7T + 83T^{2} \) |
| 89 | \( 1 - 4.81T + 89T^{2} \) |
| 97 | \( 1 - 18.8T + 97T^{2} \) |
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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
−7.83604349161059480832574464463, −7.33981934492647753266893796296, −6.31543814664365657642658070589, −5.56817012438211196551338416099, −4.74520630314494731069583131413, −4.18685098676032372855966566442, −3.73482034268868646020035134859, −2.98504025436348885875524585708, −2.15253787351089487682494582116, −0.56159669305333444463881854392,
0.56159669305333444463881854392, 2.15253787351089487682494582116, 2.98504025436348885875524585708, 3.73482034268868646020035134859, 4.18685098676032372855966566442, 4.74520630314494731069583131413, 5.56817012438211196551338416099, 6.31543814664365657642658070589, 7.33981934492647753266893796296, 7.83604349161059480832574464463
