| L(s) = 1 | − 2.31·2-s − 0.910·3-s + 3.34·4-s − 2.04·5-s + 2.10·6-s + 2.45·7-s − 3.10·8-s − 2.17·9-s + 4.72·10-s + 6.34·11-s − 3.04·12-s + 1.40·13-s − 5.67·14-s + 1.85·15-s + 0.498·16-s + 6.68·17-s + 5.01·18-s + 0.707·19-s − 6.82·20-s − 2.23·21-s − 14.6·22-s + 1.78·23-s + 2.83·24-s − 0.830·25-s − 3.25·26-s + 4.70·27-s + 8.20·28-s + ⋯ |
| L(s) = 1 | − 1.63·2-s − 0.525·3-s + 1.67·4-s − 0.913·5-s + 0.859·6-s + 0.927·7-s − 1.09·8-s − 0.723·9-s + 1.49·10-s + 1.91·11-s − 0.879·12-s + 0.389·13-s − 1.51·14-s + 0.480·15-s + 0.124·16-s + 1.62·17-s + 1.18·18-s + 0.162·19-s − 1.52·20-s − 0.487·21-s − 3.12·22-s + 0.373·23-s + 0.577·24-s − 0.166·25-s − 0.637·26-s + 0.906·27-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
| \(L(1)\) |
\(\approx\) |
\(0.6706394071\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6706394071\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.31T + 2T^{2} \) |
| 3 | \( 1 + 0.910T + 3T^{2} \) |
| 5 | \( 1 + 2.04T + 5T^{2} \) |
| 7 | \( 1 - 2.45T + 7T^{2} \) |
| 11 | \( 1 - 6.34T + 11T^{2} \) |
| 13 | \( 1 - 1.40T + 13T^{2} \) |
| 17 | \( 1 - 6.68T + 17T^{2} \) |
| 19 | \( 1 - 0.707T + 19T^{2} \) |
| 23 | \( 1 - 1.78T + 23T^{2} \) |
| 29 | \( 1 - 6.38T + 29T^{2} \) |
| 31 | \( 1 + 1.00T + 31T^{2} \) |
| 37 | \( 1 + 5.42T + 37T^{2} \) |
| 41 | \( 1 + 0.748T + 41T^{2} \) |
| 47 | \( 1 + 4.05T + 47T^{2} \) |
| 53 | \( 1 - 6.12T + 53T^{2} \) |
| 59 | \( 1 - 9.78T + 59T^{2} \) |
| 61 | \( 1 - 3.52T + 61T^{2} \) |
| 67 | \( 1 + 10.2T + 67T^{2} \) |
| 71 | \( 1 + 7.49T + 71T^{2} \) |
| 73 | \( 1 + 1.37T + 73T^{2} \) |
| 79 | \( 1 - 7.46T + 79T^{2} \) |
| 83 | \( 1 + 12.8T + 83T^{2} \) |
| 89 | \( 1 - 15.4T + 89T^{2} \) |
| 97 | \( 1 + 0.288T + 97T^{2} \) |
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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
−9.039050879064722116252777772543, −8.483672943233419878099715154254, −7.943215017209486804216277808830, −7.11190755126908631391354089293, −6.39160815144224613690575686859, −5.36926687659614577020624621480, −4.22028019642544835597761702003, −3.19782909144948252972452640835, −1.56235303906004159397483735034, −0.804356276545355559601070410399,
0.804356276545355559601070410399, 1.56235303906004159397483735034, 3.19782909144948252972452640835, 4.22028019642544835597761702003, 5.36926687659614577020624621480, 6.39160815144224613690575686859, 7.11190755126908631391354089293, 7.943215017209486804216277808830, 8.483672943233419878099715154254, 9.039050879064722116252777772543
