| L(s) = 1 | − 3-s + 4·7-s − 2·9-s − 6·11-s + 13-s + 2·19-s − 4·21-s − 23-s + 5·27-s + 9·29-s + 5·31-s + 6·33-s − 2·37-s − 39-s − 9·41-s + 4·43-s + 3·47-s + 9·49-s + 6·53-s − 2·57-s + 2·61-s − 8·63-s + 10·67-s + 69-s − 3·71-s + 7·73-s − 24·77-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.51·7-s − 2/3·9-s − 1.80·11-s + 0.277·13-s + 0.458·19-s − 0.872·21-s − 0.208·23-s + 0.962·27-s + 1.67·29-s + 0.898·31-s + 1.04·33-s − 0.328·37-s − 0.160·39-s − 1.40·41-s + 0.609·43-s + 0.437·47-s + 9/7·49-s + 0.824·53-s − 0.264·57-s + 0.256·61-s − 1.00·63-s + 1.22·67-s + 0.120·69-s − 0.356·71-s + 0.819·73-s − 2.73·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.416757087\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.416757087\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.658045597274035577262987857321, −8.263039158680734767784440545325, −7.65430632633120695676100199465, −6.63535022330794595242647107432, −5.59637642223260842167274984298, −5.14665972928625356317714150392, −4.50125749316712544034107584040, −3.05814912055831361324108561222, −2.18430524087619453430106080114, −0.790789825015232224154403917311,
0.790789825015232224154403917311, 2.18430524087619453430106080114, 3.05814912055831361324108561222, 4.50125749316712544034107584040, 5.14665972928625356317714150392, 5.59637642223260842167274984298, 6.63535022330794595242647107432, 7.65430632633120695676100199465, 8.263039158680734767784440545325, 8.658045597274035577262987857321
