| L(s) = 1 | + 3-s − 5-s − 7-s + 9-s − 4·11-s − 15-s + 5·17-s − 21-s − 23-s + 25-s + 27-s + 5·29-s − 3·31-s − 4·33-s + 35-s − 5·37-s + 3·41-s + 4·43-s − 45-s − 6·47-s − 6·49-s + 5·51-s − 3·53-s + 4·55-s − 9·59-s + 10·61-s − 63-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s − 1.20·11-s − 0.258·15-s + 1.21·17-s − 0.218·21-s − 0.208·23-s + 1/5·25-s + 0.192·27-s + 0.928·29-s − 0.538·31-s − 0.696·33-s + 0.169·35-s − 0.821·37-s + 0.468·41-s + 0.609·43-s − 0.149·45-s − 0.875·47-s − 6/7·49-s + 0.700·51-s − 0.412·53-s + 0.539·55-s − 1.17·59-s + 1.28·61-s − 0.125·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 + 7 T + p T^{2} \) |
| 73 | \( 1 + 12 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - T + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−7.889531770110116183013557221552, −7.27790368166152829731754373750, −6.45774694061894968773310551596, −5.54683276887162332841595313304, −4.89621999407984950331377691797, −3.94857238907538729478639547445, −3.17519484621868446626002038596, −2.59391978875823255419890555460, −1.37286937924532854277988188589, 0,
1.37286937924532854277988188589, 2.59391978875823255419890555460, 3.17519484621868446626002038596, 3.94857238907538729478639547445, 4.89621999407984950331377691797, 5.54683276887162332841595313304, 6.45774694061894968773310551596, 7.27790368166152829731754373750, 7.889531770110116183013557221552
