| L(s) = 1 | − 0.612·2-s − 3-s − 1.62·4-s − 1.48·5-s + 0.612·6-s + 7-s + 2.22·8-s + 9-s + 0.912·10-s − 4.13·11-s + 1.62·12-s − 6.18·13-s − 0.612·14-s + 1.48·15-s + 1.88·16-s − 0.863·17-s − 0.612·18-s + 5.94·19-s + 2.41·20-s − 21-s + 2.53·22-s − 4.86·23-s − 2.22·24-s − 2.78·25-s + 3.79·26-s − 27-s − 1.62·28-s + ⋯ |
| L(s) = 1 | − 0.433·2-s − 0.577·3-s − 0.812·4-s − 0.665·5-s + 0.250·6-s + 0.377·7-s + 0.785·8-s + 0.333·9-s + 0.288·10-s − 1.24·11-s + 0.468·12-s − 1.71·13-s − 0.163·14-s + 0.384·15-s + 0.471·16-s − 0.209·17-s − 0.144·18-s + 1.36·19-s + 0.540·20-s − 0.218·21-s + 0.540·22-s − 1.01·23-s − 0.453·24-s − 0.556·25-s + 0.743·26-s − 0.192·27-s − 0.306·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 861 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 861 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4990374648\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4990374648\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 41 | \( 1 + T \) |
| good | 2 | \( 1 + 0.612T + 2T^{2} \) |
| 5 | \( 1 + 1.48T + 5T^{2} \) |
| 11 | \( 1 + 4.13T + 11T^{2} \) |
| 13 | \( 1 + 6.18T + 13T^{2} \) |
| 17 | \( 1 + 0.863T + 17T^{2} \) |
| 19 | \( 1 - 5.94T + 19T^{2} \) |
| 23 | \( 1 + 4.86T + 23T^{2} \) |
| 29 | \( 1 - 8.25T + 29T^{2} \) |
| 31 | \( 1 - 2.32T + 31T^{2} \) |
| 37 | \( 1 - 6.60T + 37T^{2} \) |
| 43 | \( 1 - 1.51T + 43T^{2} \) |
| 47 | \( 1 + 1.43T + 47T^{2} \) |
| 53 | \( 1 - 12.7T + 53T^{2} \) |
| 59 | \( 1 + 5.15T + 59T^{2} \) |
| 61 | \( 1 - 13.5T + 61T^{2} \) |
| 67 | \( 1 - 14.7T + 67T^{2} \) |
| 71 | \( 1 - 0.561T + 71T^{2} \) |
| 73 | \( 1 - 2.94T + 73T^{2} \) |
| 79 | \( 1 - 14.8T + 79T^{2} \) |
| 83 | \( 1 + 1.17T + 83T^{2} \) |
| 89 | \( 1 - 0.339T + 89T^{2} \) |
| 97 | \( 1 + 7.48T + 97T^{2} \) |
| 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
−9.970596409865966533931885082380, −9.640447392555076365730022467489, −8.169824756479855447631822961436, −7.87955435238558959263160648498, −7.01104938430838861021660731367, −5.46276992179400699437585997739, −4.92742690781786805270998580866, −4.05261640853231440772611105598, −2.50817914450711435724629112317, −0.61093377710419326683388374037,
0.61093377710419326683388374037, 2.50817914450711435724629112317, 4.05261640853231440772611105598, 4.92742690781786805270998580866, 5.46276992179400699437585997739, 7.01104938430838861021660731367, 7.87955435238558959263160648498, 8.169824756479855447631822961436, 9.640447392555076365730022467489, 9.970596409865966533931885082380
