| L(s) = 1 | + 2-s + 3-s + 6-s − 7-s − 8-s + 9-s − 2·11-s − 14-s − 16-s + 17-s + 18-s − 21-s − 2·22-s + 23-s − 24-s + 25-s + 27-s − 31-s − 2·33-s + 34-s + 41-s − 42-s − 43-s + 46-s + 47-s − 48-s + 50-s + ⋯ |
| L(s) = 1 | + 2-s + 3-s + 6-s − 7-s − 8-s + 9-s − 2·11-s − 14-s − 16-s + 17-s + 18-s − 21-s − 2·22-s + 23-s − 24-s + 25-s + 27-s − 31-s − 2·33-s + 34-s + 41-s − 42-s − 43-s + 46-s + 47-s − 48-s + 50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 327 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 327 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.318213919\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.318213919\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 109 | \( 1 - T \) |
| good | 2 | \( 1 - T + T^{2} \) |
| 5 | \( ( 1 - T )( 1 + T ) \) |
| 7 | \( 1 + T + T^{2} \) |
| 11 | \( ( 1 + T )^{2} \) |
| 13 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( 1 - T + T^{2} \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( 1 - T + T^{2} \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( 1 + T + T^{2} \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( 1 - T + T^{2} \) |
| 43 | \( 1 + T + T^{2} \) |
| 47 | \( 1 - T + T^{2} \) |
| 53 | \( ( 1 + T )^{2} \) |
| 59 | \( 1 - T + T^{2} \) |
| 61 | \( 1 + T + T^{2} \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( 1 + T + T^{2} \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( ( 1 - T )( 1 + T ) \) |
| 97 | \( 1 + T + T^{2} \) |
| 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
−12.50500919068547497209182307711, −10.82030462196593626760549831997, −9.883541984586756496736398792138, −9.069472604374541504099987326431, −8.016272888685263920226307389618, −7.01326340171091055468225058328, −5.66785236841665618716041350791, −4.73417839756255616476753846468, −3.33050575933427173276965798930, −2.78938013482930045009737591190,
2.78938013482930045009737591190, 3.33050575933427173276965798930, 4.73417839756255616476753846468, 5.66785236841665618716041350791, 7.01326340171091055468225058328, 8.016272888685263920226307389618, 9.069472604374541504099987326431, 9.883541984586756496736398792138, 10.82030462196593626760549831997, 12.50500919068547497209182307711
