| L(s) = 1 | − 4·2-s − 26·3-s + 16·4-s − 25·5-s + 104·6-s − 22·7-s − 64·8-s + 433·9-s + 100·10-s − 768·11-s − 416·12-s − 46·13-s + 88·14-s + 650·15-s + 256·16-s + 378·17-s − 1.73e3·18-s + 1.10e3·19-s − 400·20-s + 572·21-s + 3.07e3·22-s − 1.98e3·23-s + 1.66e3·24-s + 625·25-s + 184·26-s − 4.94e3·27-s − 352·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.66·3-s + 1/2·4-s − 0.447·5-s + 1.17·6-s − 0.169·7-s − 0.353·8-s + 1.78·9-s + 0.316·10-s − 1.91·11-s − 0.833·12-s − 0.0754·13-s + 0.119·14-s + 0.745·15-s + 1/4·16-s + 0.317·17-s − 1.25·18-s + 0.699·19-s − 0.223·20-s + 0.283·21-s + 1.35·22-s − 0.782·23-s + 0.589·24-s + 1/5·25-s + 0.0533·26-s − 1.30·27-s − 0.0848·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + p^{2} T \) |
| 5 | \( 1 + p^{2} T \) |
| good | 3 | \( 1 + 26 T + p^{5} T^{2} \) |
| 7 | \( 1 + 22 T + p^{5} T^{2} \) |
| 11 | \( 1 + 768 T + p^{5} T^{2} \) |
| 13 | \( 1 + 46 T + p^{5} T^{2} \) |
| 17 | \( 1 - 378 T + p^{5} T^{2} \) |
| 19 | \( 1 - 1100 T + p^{5} T^{2} \) |
| 23 | \( 1 + 1986 T + p^{5} T^{2} \) |
| 29 | \( 1 + 5610 T + p^{5} T^{2} \) |
| 31 | \( 1 + 3988 T + p^{5} T^{2} \) |
| 37 | \( 1 + 142 T + p^{5} T^{2} \) |
| 41 | \( 1 - 1542 T + p^{5} T^{2} \) |
| 43 | \( 1 + 5026 T + p^{5} T^{2} \) |
| 47 | \( 1 - 24738 T + p^{5} T^{2} \) |
| 53 | \( 1 + 14166 T + p^{5} T^{2} \) |
| 59 | \( 1 - 28380 T + p^{5} T^{2} \) |
| 61 | \( 1 - 5522 T + p^{5} T^{2} \) |
| 67 | \( 1 + 24742 T + p^{5} T^{2} \) |
| 71 | \( 1 - 42372 T + p^{5} T^{2} \) |
| 73 | \( 1 + 52126 T + p^{5} T^{2} \) |
| 79 | \( 1 + 39640 T + p^{5} T^{2} \) |
| 83 | \( 1 + 59826 T + p^{5} T^{2} \) |
| 89 | \( 1 - 57690 T + p^{5} T^{2} \) |
| 97 | \( 1 + 144382 T + p^{5} T^{2} \) |
| 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
−18.71089443975698703059680487653, −17.90753533391967973639010814268, −16.50594414723260966299811955003, −15.67204906179363141411647710651, −12.71607735457145862734525391353, −11.33655765402030971118823252433, −10.19557438841200567182587153987, −7.52312938746834257699973811858, −5.52108618060976421966780041472, 0,
5.52108618060976421966780041472, 7.52312938746834257699973811858, 10.19557438841200567182587153987, 11.33655765402030971118823252433, 12.71607735457145862734525391353, 15.67204906179363141411647710651, 16.50594414723260966299811955003, 17.90753533391967973639010814268, 18.71089443975698703059680487653
