L(s) = 1 | + 4·2-s + 9·3-s + 16·4-s − 25·5-s + 36·6-s + 103·7-s + 64·8-s + 81·9-s − 100·10-s − 643·11-s + 144·12-s − 169·13-s + 412·14-s − 225·15-s + 256·16-s − 1.32e3·17-s + 324·18-s − 2.41e3·19-s − 400·20-s + 927·21-s − 2.57e3·22-s − 1.74e3·23-s + 576·24-s + 625·25-s − 676·26-s + 729·27-s + 1.64e3·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 0.794·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s − 1.60·11-s + 0.288·12-s − 0.277·13-s + 0.561·14-s − 0.258·15-s + 1/4·16-s − 1.11·17-s + 0.235·18-s − 1.53·19-s − 0.223·20-s + 0.458·21-s − 1.13·22-s − 0.687·23-s + 0.204·24-s + 1/5·25-s − 0.196·26-s + 0.192·27-s + 0.397·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - p^{2} T \) |
| 3 | \( 1 - p^{2} T \) |
| 5 | \( 1 + p^{2} T \) |
| 13 | \( 1 + p^{2} T \) |
good | 7 | \( 1 - 103 T + p^{5} T^{2} \) |
| 11 | \( 1 + 643 T + p^{5} T^{2} \) |
| 17 | \( 1 + 1327 T + p^{5} T^{2} \) |
| 19 | \( 1 + 2414 T + p^{5} T^{2} \) |
| 23 | \( 1 + 1745 T + p^{5} T^{2} \) |
| 29 | \( 1 - 206 p T + p^{5} T^{2} \) |
| 31 | \( 1 + 8882 T + p^{5} T^{2} \) |
| 37 | \( 1 - 8739 T + p^{5} T^{2} \) |
| 41 | \( 1 - 11909 T + p^{5} T^{2} \) |
| 43 | \( 1 + 13124 T + p^{5} T^{2} \) |
| 47 | \( 1 - 6078 T + p^{5} T^{2} \) |
| 53 | \( 1 - 16533 T + p^{5} T^{2} \) |
| 59 | \( 1 + 960 T + p^{5} T^{2} \) |
| 61 | \( 1 + 49139 T + p^{5} T^{2} \) |
| 67 | \( 1 + 14804 T + p^{5} T^{2} \) |
| 71 | \( 1 + 79359 T + p^{5} T^{2} \) |
| 73 | \( 1 + 43638 T + p^{5} T^{2} \) |
| 79 | \( 1 + 74063 T + p^{5} T^{2} \) |
| 83 | \( 1 - 98148 T + p^{5} T^{2} \) |
| 89 | \( 1 - 115951 T + p^{5} T^{2} \) |
| 97 | \( 1 - 123433 T + p^{5} T^{2} \) |
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
−10.39669473358081793914523742930, −8.879076368309046674572769493957, −8.049241666903345606129671891260, −7.37265024059842057907277923677, −6.11474489784575215752065340014, −4.83953311166796556595112986457, −4.22261080118216168652337426413, −2.78255407495985137040737943263, −1.95854356303829002884160749308, 0,
1.95854356303829002884160749308, 2.78255407495985137040737943263, 4.22261080118216168652337426413, 4.83953311166796556595112986457, 6.11474489784575215752065340014, 7.37265024059842057907277923677, 8.049241666903345606129671891260, 8.879076368309046674572769493957, 10.39669473358081793914523742930