| L(s) = 1 | − 2.79·2-s − 2.80·3-s + 5.81·4-s + 0.0114·5-s + 7.83·6-s + 2.04·7-s − 10.6·8-s + 4.84·9-s − 0.0319·10-s + 2.83·11-s − 16.2·12-s − 4.48·13-s − 5.72·14-s − 0.0319·15-s + 18.1·16-s + 0.296·17-s − 13.5·18-s − 3.81·19-s + 0.0663·20-s − 5.73·21-s − 7.91·22-s + 2.26·23-s + 29.8·24-s − 4.99·25-s + 12.5·26-s − 5.17·27-s + 11.8·28-s + ⋯ |
| L(s) = 1 | − 1.97·2-s − 1.61·3-s + 2.90·4-s + 0.00510·5-s + 3.19·6-s + 0.773·7-s − 3.77·8-s + 1.61·9-s − 0.0100·10-s + 0.853·11-s − 4.70·12-s − 1.24·13-s − 1.52·14-s − 0.00825·15-s + 4.54·16-s + 0.0719·17-s − 3.19·18-s − 0.875·19-s + 0.0148·20-s − 1.25·21-s − 1.68·22-s + 0.471·23-s + 6.09·24-s − 0.999·25-s + 2.46·26-s − 0.996·27-s + 2.24·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8011 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8011 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1064015210\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1064015210\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 8011 | \( 1+O(T) \) |
| good | 2 | \( 1 + 2.79T + 2T^{2} \) |
| 3 | \( 1 + 2.80T + 3T^{2} \) |
| 5 | \( 1 - 0.0114T + 5T^{2} \) |
| 7 | \( 1 - 2.04T + 7T^{2} \) |
| 11 | \( 1 - 2.83T + 11T^{2} \) |
| 13 | \( 1 + 4.48T + 13T^{2} \) |
| 17 | \( 1 - 0.296T + 17T^{2} \) |
| 19 | \( 1 + 3.81T + 19T^{2} \) |
| 23 | \( 1 - 2.26T + 23T^{2} \) |
| 29 | \( 1 + 8.53T + 29T^{2} \) |
| 31 | \( 1 + 5.07T + 31T^{2} \) |
| 37 | \( 1 + 4.77T + 37T^{2} \) |
| 41 | \( 1 + 4.60T + 41T^{2} \) |
| 43 | \( 1 + 8.56T + 43T^{2} \) |
| 47 | \( 1 + 8.81T + 47T^{2} \) |
| 53 | \( 1 - 2.70T + 53T^{2} \) |
| 59 | \( 1 - 4.17T + 59T^{2} \) |
| 61 | \( 1 + 9.11T + 61T^{2} \) |
| 67 | \( 1 + 10.3T + 67T^{2} \) |
| 71 | \( 1 + 7.07T + 71T^{2} \) |
| 73 | \( 1 + 16.4T + 73T^{2} \) |
| 79 | \( 1 - 10.4T + 79T^{2} \) |
| 83 | \( 1 + 11.1T + 83T^{2} \) |
| 89 | \( 1 + 10.5T + 89T^{2} \) |
| 97 | \( 1 + 7.21T + 97T^{2} \) |
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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
−7.73807126629502734724479062759, −7.18478925155152123861671265213, −6.73736626836864595490012216383, −5.95940113836988479237193763713, −5.41004554004407510141689771187, −4.50085583780048397491806403871, −3.27607087523332392518634416949, −1.78804420144499204030218225941, −1.65859566055504896742606517565, −0.23809001114511088441700622741,
0.23809001114511088441700622741, 1.65859566055504896742606517565, 1.78804420144499204030218225941, 3.27607087523332392518634416949, 4.50085583780048397491806403871, 5.41004554004407510141689771187, 5.95940113836988479237193763713, 6.73736626836864595490012216383, 7.18478925155152123861671265213, 7.73807126629502734724479062759
