| L(s) = 1 | − 2-s − 3.08·3-s + 4-s − 2.41·5-s + 3.08·6-s + 1.43·7-s − 8-s + 6.49·9-s + 2.41·10-s − 11-s − 3.08·12-s − 4.84·13-s − 1.43·14-s + 7.44·15-s + 16-s + 6.84·17-s − 6.49·18-s − 8.01·19-s − 2.41·20-s − 4.41·21-s + 22-s − 2.16·23-s + 3.08·24-s + 0.836·25-s + 4.84·26-s − 10.7·27-s + 1.43·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.77·3-s + 0.5·4-s − 1.08·5-s + 1.25·6-s + 0.541·7-s − 0.353·8-s + 2.16·9-s + 0.763·10-s − 0.301·11-s − 0.889·12-s − 1.34·13-s − 0.382·14-s + 1.92·15-s + 0.250·16-s + 1.66·17-s − 1.53·18-s − 1.83·19-s − 0.540·20-s − 0.963·21-s + 0.213·22-s − 0.451·23-s + 0.629·24-s + 0.167·25-s + 0.950·26-s − 2.07·27-s + 0.270·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1342 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1342 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2585897494\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2585897494\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 61 | \( 1 - T \) |
| good | 3 | \( 1 + 3.08T + 3T^{2} \) |
| 5 | \( 1 + 2.41T + 5T^{2} \) |
| 7 | \( 1 - 1.43T + 7T^{2} \) |
| 13 | \( 1 + 4.84T + 13T^{2} \) |
| 17 | \( 1 - 6.84T + 17T^{2} \) |
| 19 | \( 1 + 8.01T + 19T^{2} \) |
| 23 | \( 1 + 2.16T + 23T^{2} \) |
| 29 | \( 1 - 0.718T + 29T^{2} \) |
| 31 | \( 1 + 6.21T + 31T^{2} \) |
| 37 | \( 1 + 6.04T + 37T^{2} \) |
| 41 | \( 1 + 12.3T + 41T^{2} \) |
| 43 | \( 1 - 9.91T + 43T^{2} \) |
| 47 | \( 1 - 1.76T + 47T^{2} \) |
| 53 | \( 1 - 10.1T + 53T^{2} \) |
| 59 | \( 1 + 5.99T + 59T^{2} \) |
| 67 | \( 1 + 1.18T + 67T^{2} \) |
| 71 | \( 1 - 5.94T + 71T^{2} \) |
| 73 | \( 1 - 11.7T + 73T^{2} \) |
| 79 | \( 1 + 13.6T + 79T^{2} \) |
| 83 | \( 1 + 3.04T + 83T^{2} \) |
| 89 | \( 1 - 18.2T + 89T^{2} \) |
| 97 | \( 1 - 2.94T + 97T^{2} \) |
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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
−10.06735983264086548788353351580, −8.723290127992264765667254380876, −7.70409286838634327287684129327, −7.35470830138969636464892867447, −6.38028407883421432228121358077, −5.43145236337942048551774577934, −4.75291846826111808895011973449, −3.72834612923334772011197452325, −1.92761440774831097599338878342, −0.43743090127751653323572692254,
0.43743090127751653323572692254, 1.92761440774831097599338878342, 3.72834612923334772011197452325, 4.75291846826111808895011973449, 5.43145236337942048551774577934, 6.38028407883421432228121358077, 7.35470830138969636464892867447, 7.70409286838634327287684129327, 8.723290127992264765667254380876, 10.06735983264086548788353351580
