| L(s) = 1 | + 0.256·2-s − 1.93·4-s − 2.52·5-s + 4.03·7-s − 1.00·8-s − 0.648·10-s − 2.05·11-s − 3.92·13-s + 1.03·14-s + 3.60·16-s − 7.80·17-s − 6.21·19-s + 4.88·20-s − 0.526·22-s − 5.06·23-s + 1.38·25-s − 1.00·26-s − 7.80·28-s − 7.27·29-s + 2.94·32-s − 2.00·34-s − 10.2·35-s + 1.56·37-s − 1.59·38-s + 2.55·40-s + 5.69·41-s − 2.05·43-s + ⋯ |
| L(s) = 1 | + 0.181·2-s − 0.967·4-s − 1.13·5-s + 1.52·7-s − 0.356·8-s − 0.205·10-s − 0.618·11-s − 1.08·13-s + 0.276·14-s + 0.902·16-s − 1.89·17-s − 1.42·19-s + 1.09·20-s − 0.112·22-s − 1.05·23-s + 0.277·25-s − 0.197·26-s − 1.47·28-s − 1.35·29-s + 0.520·32-s − 0.343·34-s − 1.72·35-s + 0.257·37-s − 0.258·38-s + 0.403·40-s + 0.888·41-s − 0.313·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8649 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8649 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3411743843\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3411743843\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 31 | \( 1 \) |
| good | 2 | \( 1 - 0.256T + 2T^{2} \) |
| 5 | \( 1 + 2.52T + 5T^{2} \) |
| 7 | \( 1 - 4.03T + 7T^{2} \) |
| 11 | \( 1 + 2.05T + 11T^{2} \) |
| 13 | \( 1 + 3.92T + 13T^{2} \) |
| 17 | \( 1 + 7.80T + 17T^{2} \) |
| 19 | \( 1 + 6.21T + 19T^{2} \) |
| 23 | \( 1 + 5.06T + 23T^{2} \) |
| 29 | \( 1 + 7.27T + 29T^{2} \) |
| 37 | \( 1 - 1.56T + 37T^{2} \) |
| 41 | \( 1 - 5.69T + 41T^{2} \) |
| 43 | \( 1 + 2.05T + 43T^{2} \) |
| 47 | \( 1 - 5.10T + 47T^{2} \) |
| 53 | \( 1 + 1.34T + 53T^{2} \) |
| 59 | \( 1 + 4.66T + 59T^{2} \) |
| 61 | \( 1 + 7.65T + 61T^{2} \) |
| 67 | \( 1 - 0.149T + 67T^{2} \) |
| 71 | \( 1 - 7.77T + 71T^{2} \) |
| 73 | \( 1 - 8.38T + 73T^{2} \) |
| 79 | \( 1 - 0.852T + 79T^{2} \) |
| 83 | \( 1 - 9.98T + 83T^{2} \) |
| 89 | \( 1 + 10.1T + 89T^{2} \) |
| 97 | \( 1 + 12.0T + 97T^{2} \) |
| 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
−7.921621225013170373840478298588, −7.37276062380816065700370192298, −6.37684845634700602448641846966, −5.41914617235268346073736372291, −4.79752008977210808447173887860, −4.22261289738600265433107389015, −3.97721089109130265909205498658, −2.54986885554534781635281327813, −1.86674391043310039207101053935, −0.26599091315136702220662811627,
0.26599091315136702220662811627, 1.86674391043310039207101053935, 2.54986885554534781635281327813, 3.97721089109130265909205498658, 4.22261289738600265433107389015, 4.79752008977210808447173887860, 5.41914617235268346073736372291, 6.37684845634700602448641846966, 7.37276062380816065700370192298, 7.921621225013170373840478298588
