| L(s) = 1 | − 3.16·3-s − 1.06·5-s − 0.836·7-s + 7.02·9-s − 3.32·11-s − 1.60·13-s + 3.37·15-s + 17-s − 6.50·19-s + 2.64·21-s − 1.48·23-s − 3.86·25-s − 12.7·27-s + 7.50·29-s − 4.45·31-s + 10.5·33-s + 0.891·35-s − 3.56·37-s + 5.08·39-s + 4.62·41-s − 4.37·43-s − 7.49·45-s + 2.49·47-s − 6.30·49-s − 3.16·51-s − 12.6·53-s + 3.54·55-s + ⋯ |
| L(s) = 1 | − 1.82·3-s − 0.477·5-s − 0.315·7-s + 2.34·9-s − 1.00·11-s − 0.445·13-s + 0.872·15-s + 0.242·17-s − 1.49·19-s + 0.577·21-s − 0.310·23-s − 0.772·25-s − 2.45·27-s + 1.39·29-s − 0.800·31-s + 1.83·33-s + 0.150·35-s − 0.585·37-s + 0.813·39-s + 0.722·41-s − 0.667·43-s − 1.11·45-s + 0.364·47-s − 0.900·49-s − 0.443·51-s − 1.74·53-s + 0.478·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1004169440\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1004169440\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 - T \) |
| 59 | \( 1 - T \) |
| good | 3 | \( 1 + 3.16T + 3T^{2} \) |
| 5 | \( 1 + 1.06T + 5T^{2} \) |
| 7 | \( 1 + 0.836T + 7T^{2} \) |
| 11 | \( 1 + 3.32T + 11T^{2} \) |
| 13 | \( 1 + 1.60T + 13T^{2} \) |
| 19 | \( 1 + 6.50T + 19T^{2} \) |
| 23 | \( 1 + 1.48T + 23T^{2} \) |
| 29 | \( 1 - 7.50T + 29T^{2} \) |
| 31 | \( 1 + 4.45T + 31T^{2} \) |
| 37 | \( 1 + 3.56T + 37T^{2} \) |
| 41 | \( 1 - 4.62T + 41T^{2} \) |
| 43 | \( 1 + 4.37T + 43T^{2} \) |
| 47 | \( 1 - 2.49T + 47T^{2} \) |
| 53 | \( 1 + 12.6T + 53T^{2} \) |
| 61 | \( 1 - 1.68T + 61T^{2} \) |
| 67 | \( 1 + 3.14T + 67T^{2} \) |
| 71 | \( 1 - 1.93T + 71T^{2} \) |
| 73 | \( 1 - 1.89T + 73T^{2} \) |
| 79 | \( 1 - 0.220T + 79T^{2} \) |
| 83 | \( 1 + 5.92T + 83T^{2} \) |
| 89 | \( 1 - 2.30T + 89T^{2} \) |
| 97 | \( 1 + 0.988T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.76652943922760632376412738744, −6.88588084987092029552963595982, −6.44065232672802025837884990583, −5.74025246272267780848480321578, −5.09633125819704421023086005262, −4.50281085026806418478754786858, −3.79674125710771273007612444932, −2.60524132216227146007069451034, −1.51180010297634566302604876142, −0.17397206039755972514383928642,
0.17397206039755972514383928642, 1.51180010297634566302604876142, 2.60524132216227146007069451034, 3.79674125710771273007612444932, 4.50281085026806418478754786858, 5.09633125819704421023086005262, 5.74025246272267780848480321578, 6.44065232672802025837884990583, 6.88588084987092029552963595982, 7.76652943922760632376412738744
