| L(s) = 1 | + 5.54·2-s + 22.7·4-s + 5·5-s + 17.7·7-s + 81.4·8-s + 27.7·10-s + 11·11-s − 53.5·13-s + 98.2·14-s + 269.·16-s − 112.·17-s − 1.24·19-s + 113.·20-s + 60.9·22-s + 78.4·23-s + 25·25-s − 296.·26-s + 402.·28-s − 174.·29-s + 82.5·31-s + 842.·32-s − 622.·34-s + 88.6·35-s − 149.·37-s − 6.89·38-s + 407.·40-s − 414.·41-s + ⋯ |
| L(s) = 1 | + 1.95·2-s + 2.83·4-s + 0.447·5-s + 0.957·7-s + 3.60·8-s + 0.876·10-s + 0.301·11-s − 1.14·13-s + 1.87·14-s + 4.21·16-s − 1.60·17-s − 0.0150·19-s + 1.26·20-s + 0.590·22-s + 0.710·23-s + 0.200·25-s − 2.23·26-s + 2.71·28-s − 1.11·29-s + 0.478·31-s + 4.65·32-s − 3.14·34-s + 0.428·35-s − 0.662·37-s − 0.0294·38-s + 1.60·40-s − 1.57·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(8.039555192\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.039555192\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| 11 | \( 1 - 11T \) |
| good | 2 | \( 1 - 5.54T + 8T^{2} \) |
| 7 | \( 1 - 17.7T + 343T^{2} \) |
| 13 | \( 1 + 53.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 112.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 1.24T + 6.85e3T^{2} \) |
| 23 | \( 1 - 78.4T + 1.21e4T^{2} \) |
| 29 | \( 1 + 174.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 82.5T + 2.97e4T^{2} \) |
| 37 | \( 1 + 149.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 414.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 182.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 438.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 490.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 6.02T + 2.05e5T^{2} \) |
| 61 | \( 1 - 434.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 935.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 510.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.04e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 226.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.18e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.44e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 825.T + 9.12e5T^{2} \) |
| 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
−11.02376634239318693754219747442, −10.02851522814304760201630649439, −8.555801790984331616115559847748, −7.23974038578868916546597329164, −6.70705525141567081109518883477, −5.45840693214865710957952068088, −4.87450429889828937077592786546, −3.97973928712026310674907229711, −2.58088430699331823546332987798, −1.76018492814994346116077475606,
1.76018492814994346116077475606, 2.58088430699331823546332987798, 3.97973928712026310674907229711, 4.87450429889828937077592786546, 5.45840693214865710957952068088, 6.70705525141567081109518883477, 7.23974038578868916546597329164, 8.555801790984331616115559847748, 10.02851522814304760201630649439, 11.02376634239318693754219747442
