| L(s) = 1 | − 3.02·3-s − 0.369·7-s + 6.15·9-s + 1.74·11-s + 1.11·13-s + 5.48·17-s + 3.75·19-s + 1.11·21-s − 7.24·23-s − 9.54·27-s + 4.19·29-s − 0.305·31-s − 5.28·33-s − 9.21·37-s − 3.37·39-s − 4.18·41-s − 7.17·43-s − 0.810·47-s − 6.86·49-s − 16.6·51-s − 3.91·53-s − 11.3·57-s + 1.85·59-s + 9.68·61-s − 2.27·63-s + 12.4·67-s + 21.9·69-s + ⋯ |
| L(s) = 1 | − 1.74·3-s − 0.139·7-s + 2.05·9-s + 0.526·11-s + 0.309·13-s + 1.33·17-s + 0.860·19-s + 0.244·21-s − 1.51·23-s − 1.83·27-s + 0.778·29-s − 0.0549·31-s − 0.919·33-s − 1.51·37-s − 0.540·39-s − 0.653·41-s − 1.09·43-s − 0.118·47-s − 0.980·49-s − 2.32·51-s − 0.537·53-s − 1.50·57-s + 0.241·59-s + 1.24·61-s − 0.286·63-s + 1.52·67-s + 2.63·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 3.02T + 3T^{2} \) |
| 7 | \( 1 + 0.369T + 7T^{2} \) |
| 11 | \( 1 - 1.74T + 11T^{2} \) |
| 13 | \( 1 - 1.11T + 13T^{2} \) |
| 17 | \( 1 - 5.48T + 17T^{2} \) |
| 19 | \( 1 - 3.75T + 19T^{2} \) |
| 23 | \( 1 + 7.24T + 23T^{2} \) |
| 29 | \( 1 - 4.19T + 29T^{2} \) |
| 31 | \( 1 + 0.305T + 31T^{2} \) |
| 37 | \( 1 + 9.21T + 37T^{2} \) |
| 41 | \( 1 + 4.18T + 41T^{2} \) |
| 43 | \( 1 + 7.17T + 43T^{2} \) |
| 47 | \( 1 + 0.810T + 47T^{2} \) |
| 53 | \( 1 + 3.91T + 53T^{2} \) |
| 59 | \( 1 - 1.85T + 59T^{2} \) |
| 61 | \( 1 - 9.68T + 61T^{2} \) |
| 67 | \( 1 - 12.4T + 67T^{2} \) |
| 71 | \( 1 + 11.6T + 71T^{2} \) |
| 73 | \( 1 - 3.43T + 73T^{2} \) |
| 79 | \( 1 + 5.69T + 79T^{2} \) |
| 83 | \( 1 + 7.13T + 83T^{2} \) |
| 89 | \( 1 - 1.52T + 89T^{2} \) |
| 97 | \( 1 - 6.39T + 97T^{2} \) |
| 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
−7.00364057993219215496535318469, −6.59311944377367667593444922721, −5.87131237073132573844281550045, −5.38602905350792804592498480954, −4.79438228771293948050335922112, −3.88468236994066049650940188208, −3.25803285537882977116847336630, −1.76137207122073213085962079858, −1.06547929116257750299522512037, 0,
1.06547929116257750299522512037, 1.76137207122073213085962079858, 3.25803285537882977116847336630, 3.88468236994066049650940188208, 4.79438228771293948050335922112, 5.38602905350792804592498480954, 5.87131237073132573844281550045, 6.59311944377367667593444922721, 7.00364057993219215496535318469
