| L(s) = 1 | + 1.62·2-s − 2.30·3-s + 0.637·4-s + 3.05·5-s − 3.74·6-s − 2.09·7-s − 2.21·8-s + 2.31·9-s + 4.95·10-s + 4.62·11-s − 1.46·12-s + 1.50·13-s − 3.40·14-s − 7.03·15-s − 4.86·16-s − 5.03·17-s + 3.75·18-s + 0.751·19-s + 1.94·20-s + 4.83·21-s + 7.51·22-s + 9.23·23-s + 5.10·24-s + 4.31·25-s + 2.43·26-s + 1.58·27-s − 1.33·28-s + ⋯ |
| L(s) = 1 | + 1.14·2-s − 1.33·3-s + 0.318·4-s + 1.36·5-s − 1.52·6-s − 0.793·7-s − 0.782·8-s + 0.771·9-s + 1.56·10-s + 1.39·11-s − 0.423·12-s + 0.416·13-s − 0.911·14-s − 1.81·15-s − 1.21·16-s − 1.22·17-s + 0.885·18-s + 0.172·19-s + 0.434·20-s + 1.05·21-s + 1.60·22-s + 1.92·23-s + 1.04·24-s + 0.862·25-s + 0.478·26-s + 0.304·27-s − 0.252·28-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
| \(L(1)\) |
\(\approx\) |
\(2.189687491\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.189687491\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.62T + 2T^{2} \) |
| 3 | \( 1 + 2.30T + 3T^{2} \) |
| 5 | \( 1 - 3.05T + 5T^{2} \) |
| 7 | \( 1 + 2.09T + 7T^{2} \) |
| 11 | \( 1 - 4.62T + 11T^{2} \) |
| 13 | \( 1 - 1.50T + 13T^{2} \) |
| 17 | \( 1 + 5.03T + 17T^{2} \) |
| 19 | \( 1 - 0.751T + 19T^{2} \) |
| 23 | \( 1 - 9.23T + 23T^{2} \) |
| 29 | \( 1 + 1.87T + 29T^{2} \) |
| 31 | \( 1 - 0.331T + 31T^{2} \) |
| 37 | \( 1 - 3.45T + 37T^{2} \) |
| 41 | \( 1 - 7.55T + 41T^{2} \) |
| 47 | \( 1 - 4.83T + 47T^{2} \) |
| 53 | \( 1 + 5.28T + 53T^{2} \) |
| 59 | \( 1 - 9.03T + 59T^{2} \) |
| 61 | \( 1 - 11.4T + 61T^{2} \) |
| 67 | \( 1 - 2.94T + 67T^{2} \) |
| 71 | \( 1 - 0.0292T + 71T^{2} \) |
| 73 | \( 1 + 2.78T + 73T^{2} \) |
| 79 | \( 1 - 6.68T + 79T^{2} \) |
| 83 | \( 1 - 14.4T + 83T^{2} \) |
| 89 | \( 1 + 3.88T + 89T^{2} \) |
| 97 | \( 1 + 1.21T + 97T^{2} \) |
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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
−9.208575146415807491844383591289, −8.943997941117889607293087114054, −6.89945734026685074847940458043, −6.46839387255105300603601057996, −6.00982761573304649948781832740, −5.29296724489309798166534294668, −4.55118996380536766442485292401, −3.57640491272864867334072263670, −2.42781243741153525325318596312, −0.928811891557264616366562618200,
0.928811891557264616366562618200, 2.42781243741153525325318596312, 3.57640491272864867334072263670, 4.55118996380536766442485292401, 5.29296724489309798166534294668, 6.00982761573304649948781832740, 6.46839387255105300603601057996, 6.89945734026685074847940458043, 8.943997941117889607293087114054, 9.208575146415807491844383591289
