| L(s) = 1 | + 8.36·2-s − 9·3-s + 37.9·4-s − 75.2·6-s − 15.4·7-s + 49.6·8-s + 81·9-s − 121·11-s − 341.·12-s + 624.·13-s − 129.·14-s − 798.·16-s + 1.25e3·17-s + 677.·18-s − 440.·19-s + 138.·21-s − 1.01e3·22-s + 2.53e3·23-s − 446.·24-s + 5.22e3·26-s − 729·27-s − 585.·28-s − 3.65e3·29-s − 8.00e3·31-s − 8.26e3·32-s + 1.08e3·33-s + 1.04e4·34-s + ⋯ |
| L(s) = 1 | + 1.47·2-s − 0.577·3-s + 1.18·4-s − 0.853·6-s − 0.119·7-s + 0.274·8-s + 0.333·9-s − 0.301·11-s − 0.684·12-s + 1.02·13-s − 0.176·14-s − 0.780·16-s + 1.05·17-s + 0.492·18-s − 0.279·19-s + 0.0687·21-s − 0.445·22-s + 0.998·23-s − 0.158·24-s + 1.51·26-s − 0.192·27-s − 0.141·28-s − 0.806·29-s − 1.49·31-s − 1.42·32-s + 0.174·33-s + 1.55·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(4.152014381\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.152014381\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 9T \) |
| 5 | \( 1 \) |
| 11 | \( 1 + 121T \) |
| good | 2 | \( 1 - 8.36T + 32T^{2} \) |
| 7 | \( 1 + 15.4T + 1.68e4T^{2} \) |
| 13 | \( 1 - 624.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.25e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 440.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 2.53e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 3.65e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 8.00e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.20e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.32e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 4.65e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.29e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.38e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 4.64e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 5.59e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.06e3T + 1.35e9T^{2} \) |
| 71 | \( 1 - 6.25e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 6.08e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 5.38e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 5.35e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 3.38e3T + 5.58e9T^{2} \) |
| 97 | \( 1 - 3.02e4T + 8.58e9T^{2} \) |
| 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
−9.569941112119286135522374947483, −8.615459619425008304995493106160, −7.35780623072521411782596052686, −6.60503692101893645253158161234, −5.52858544417441578111448505963, −5.34500810197405329966102069562, −3.99207671484462203538640404725, −3.44965101974111980330366583631, −2.15634080463039490484642105593, −0.75955284656520670844282224019,
0.75955284656520670844282224019, 2.15634080463039490484642105593, 3.44965101974111980330366583631, 3.99207671484462203538640404725, 5.34500810197405329966102069562, 5.52858544417441578111448505963, 6.60503692101893645253158161234, 7.35780623072521411782596052686, 8.615459619425008304995493106160, 9.569941112119286135522374947483
