| L(s) = 1 | − 0.369·2-s + 0.00229·3-s − 1.86·4-s − 2.37·5-s − 0.000846·6-s + 4.42·7-s + 1.42·8-s − 2.99·9-s + 0.876·10-s + 4.37·11-s − 0.00427·12-s − 4.15·13-s − 1.63·14-s − 0.00544·15-s + 3.20·16-s + 2.13·17-s + 1.10·18-s − 1.88·19-s + 4.42·20-s + 0.0101·21-s − 1.61·22-s − 1.32·23-s + 0.00326·24-s + 0.645·25-s + 1.53·26-s − 0.0137·27-s − 8.24·28-s + ⋯ |
| L(s) = 1 | − 0.260·2-s + 0.00132·3-s − 0.931·4-s − 1.06·5-s − 0.000345·6-s + 1.67·7-s + 0.504·8-s − 0.999·9-s + 0.277·10-s + 1.32·11-s − 0.00123·12-s − 1.15·13-s − 0.436·14-s − 0.00140·15-s + 0.800·16-s + 0.518·17-s + 0.260·18-s − 0.431·19-s + 0.990·20-s + 0.00221·21-s − 0.344·22-s − 0.276·23-s + 0.000667·24-s + 0.129·25-s + 0.300·26-s − 0.00264·27-s − 1.55·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9914599668\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9914599668\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 \) |
| good | 2 | \( 1 + 0.369T + 2T^{2} \) |
| 3 | \( 1 - 0.00229T + 3T^{2} \) |
| 5 | \( 1 + 2.37T + 5T^{2} \) |
| 7 | \( 1 - 4.42T + 7T^{2} \) |
| 11 | \( 1 - 4.37T + 11T^{2} \) |
| 13 | \( 1 + 4.15T + 13T^{2} \) |
| 17 | \( 1 - 2.13T + 17T^{2} \) |
| 19 | \( 1 + 1.88T + 19T^{2} \) |
| 23 | \( 1 + 1.32T + 23T^{2} \) |
| 29 | \( 1 + 7.49T + 29T^{2} \) |
| 31 | \( 1 - 2.12T + 31T^{2} \) |
| 37 | \( 1 + 2.46T + 37T^{2} \) |
| 41 | \( 1 - 8.51T + 41T^{2} \) |
| 47 | \( 1 - 8.63T + 47T^{2} \) |
| 53 | \( 1 + 0.533T + 53T^{2} \) |
| 59 | \( 1 - 0.466T + 59T^{2} \) |
| 61 | \( 1 - 8.94T + 61T^{2} \) |
| 67 | \( 1 - 6.26T + 67T^{2} \) |
| 71 | \( 1 - 1.94T + 71T^{2} \) |
| 73 | \( 1 - 1.11T + 73T^{2} \) |
| 79 | \( 1 - 15.8T + 79T^{2} \) |
| 83 | \( 1 + 12.5T + 83T^{2} \) |
| 89 | \( 1 + 4.09T + 89T^{2} \) |
| 97 | \( 1 - 0.625T + 97T^{2} \) |
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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
−9.017246955835858250168577338209, −8.469000395790382929661273791010, −7.79439264663050889253081019598, −7.31131444695865761331149708428, −5.83378126050860261820235363666, −5.03241280584571539154663684711, −4.27957913518989278622818176882, −3.65844137915707530344347666568, −2.07879126774555052190443045297, −0.71718575461662195376188889993,
0.71718575461662195376188889993, 2.07879126774555052190443045297, 3.65844137915707530344347666568, 4.27957913518989278622818176882, 5.03241280584571539154663684711, 5.83378126050860261820235363666, 7.31131444695865761331149708428, 7.79439264663050889253081019598, 8.469000395790382929661273791010, 9.017246955835858250168577338209
