| L(s) = 1 | + 4·2-s + 9·3-s + 16·4-s − 54·5-s + 36·6-s + 49·7-s + 64·8-s + 81·9-s − 216·10-s − 192·11-s + 144·12-s + 169·13-s + 196·14-s − 486·15-s + 256·16-s − 1.42e3·17-s + 324·18-s + 1.74e3·19-s − 864·20-s + 441·21-s − 768·22-s − 3.79e3·23-s + 576·24-s − 209·25-s + 676·26-s + 729·27-s + 784·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.965·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.683·10-s − 0.478·11-s + 0.288·12-s + 0.277·13-s + 0.267·14-s − 0.557·15-s + 1/4·16-s − 1.19·17-s + 0.235·18-s + 1.11·19-s − 0.482·20-s + 0.218·21-s − 0.338·22-s − 1.49·23-s + 0.204·24-s − 0.0668·25-s + 0.196·26-s + 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - p^{2} T \) |
| 7 | \( 1 - p^{2} T \) |
| 13 | \( 1 - p^{2} T \) |
| good | 5 | \( 1 + 54 T + p^{5} T^{2} \) |
| 11 | \( 1 + 192 T + p^{5} T^{2} \) |
| 17 | \( 1 + 1422 T + p^{5} T^{2} \) |
| 19 | \( 1 - 92 p T + p^{5} T^{2} \) |
| 23 | \( 1 + 3792 T + p^{5} T^{2} \) |
| 29 | \( 1 + 954 T + p^{5} T^{2} \) |
| 31 | \( 1 + 568 T + p^{5} T^{2} \) |
| 37 | \( 1 - 7886 T + p^{5} T^{2} \) |
| 41 | \( 1 + 14802 T + p^{5} T^{2} \) |
| 43 | \( 1 - 4964 T + p^{5} T^{2} \) |
| 47 | \( 1 + 18948 T + p^{5} T^{2} \) |
| 53 | \( 1 + 426 T + p^{5} T^{2} \) |
| 59 | \( 1 - 34872 T + p^{5} T^{2} \) |
| 61 | \( 1 + 25618 T + p^{5} T^{2} \) |
| 67 | \( 1 + 67060 T + p^{5} T^{2} \) |
| 71 | \( 1 + 28428 T + p^{5} T^{2} \) |
| 73 | \( 1 + 22894 T + p^{5} T^{2} \) |
| 79 | \( 1 + 1408 T + p^{5} T^{2} \) |
| 83 | \( 1 + 17304 T + p^{5} T^{2} \) |
| 89 | \( 1 + 93690 T + p^{5} T^{2} \) |
| 97 | \( 1 - 16826 T + p^{5} T^{2} \) |
| 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
−9.588799646227510532617139795388, −8.389873407751226519244235193178, −7.81727537196367218568248922072, −6.96133797171718847317187171148, −5.76391815936787233906114017714, −4.59495417357038157242404046543, −3.88047110823288346725278681275, −2.86004156254573473647139346450, −1.67238916731038755293690425631, 0,
1.67238916731038755293690425631, 2.86004156254573473647139346450, 3.88047110823288346725278681275, 4.59495417357038157242404046543, 5.76391815936787233906114017714, 6.96133797171718847317187171148, 7.81727537196367218568248922072, 8.389873407751226519244235193178, 9.588799646227510532617139795388
