| 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
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6173250271\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6173250271\) |
| \(L(1)\) |
|
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
−11.74650745927727950131795816314, −10.45945189387656472092812801547, −9.624193688403898187388115849847, −9.045224346074777022319737595946, −8.467812092544809563656608280323, −7.14944691097300723106608156832, −6.50100474412380231900694567502, −4.42386325006468296262737482580, −3.48379135769085281071487237739, −1.70458306709519005726191879779,
1.70458306709519005726191879779, 3.48379135769085281071487237739, 4.42386325006468296262737482580, 6.50100474412380231900694567502, 7.14944691097300723106608156832, 8.467812092544809563656608280323, 9.045224346074777022319737595946, 9.624193688403898187388115849847, 10.45945189387656472092812801547, 11.74650745927727950131795816314
