| L(s) = 1 | − 3-s − 5-s + 7-s + 9-s − 11-s − 4·13-s + 15-s + 17-s + 3·19-s − 21-s − 23-s + 25-s − 27-s − 3·29-s − 2·31-s + 33-s − 35-s + 8·37-s + 4·39-s − 2·41-s − 43-s − 45-s + 2·47-s + 49-s − 51-s + 11·53-s + 55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.377·7-s + 1/3·9-s − 0.301·11-s − 1.10·13-s + 0.258·15-s + 0.242·17-s + 0.688·19-s − 0.218·21-s − 0.208·23-s + 1/5·25-s − 0.192·27-s − 0.557·29-s − 0.359·31-s + 0.174·33-s − 0.169·35-s + 1.31·37-s + 0.640·39-s − 0.312·41-s − 0.152·43-s − 0.149·45-s + 0.291·47-s + 1/7·49-s − 0.140·51-s + 1.51·53-s + 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| good | 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 11 T + p T^{2} \) |
| 59 | \( 1 - 13 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 - 11 T + p T^{2} \) |
| 97 | \( 1 + 11 T + p T^{2} \) |
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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
−7.41219653634370167243245488713, −6.85561641267194273263816429658, −5.79576232564378428907849581486, −5.39286415644365016990793571836, −4.60641058105008417328865273951, −4.01569604157488650304870245107, −3.01795710713111506463150051733, −2.18847755127359144598013416701, −1.08006460552862516929581830062, 0,
1.08006460552862516929581830062, 2.18847755127359144598013416701, 3.01795710713111506463150051733, 4.01569604157488650304870245107, 4.60641058105008417328865273951, 5.39286415644365016990793571836, 5.79576232564378428907849581486, 6.85561641267194273263816429658, 7.41219653634370167243245488713
