| L(s) = 1 | − 3-s + 7-s + 9-s − 4·13-s − 6·17-s − 2·19-s − 21-s − 9·23-s − 27-s + 6·29-s − 2·31-s + 2·37-s + 4·39-s + 6·41-s − 43-s + 9·47-s − 6·49-s + 6·51-s − 6·53-s + 2·57-s + 12·59-s + 2·61-s + 63-s − 5·67-s + 9·69-s + 12·71-s − 4·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.377·7-s + 1/3·9-s − 1.10·13-s − 1.45·17-s − 0.458·19-s − 0.218·21-s − 1.87·23-s − 0.192·27-s + 1.11·29-s − 0.359·31-s + 0.328·37-s + 0.640·39-s + 0.937·41-s − 0.152·43-s + 1.31·47-s − 6/7·49-s + 0.840·51-s − 0.824·53-s + 0.264·57-s + 1.56·59-s + 0.256·61-s + 0.125·63-s − 0.610·67-s + 1.08·69-s + 1.42·71-s − 0.468·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 43 | \( 1 + T \) |
| good | 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−14.57979846338424, −14.34716766827213, −13.76610331134312, −13.07876346249368, −12.68603730360306, −12.11154268153246, −11.69999347601947, −11.18858135017921, −10.64725377836869, −10.13292876349413, −9.672195924872420, −9.052815520209135, −8.380510619368602, −7.969246411007359, −7.252853236762221, −6.811448049397319, −6.159848659322155, −5.723579033469881, −4.959205615373227, −4.373680325266646, −4.165555300544674, −3.100262118887869, −2.181275575319050, −2.003894771875656, −0.7720637560017615, 0,
0.7720637560017615, 2.003894771875656, 2.181275575319050, 3.100262118887869, 4.165555300544674, 4.373680325266646, 4.959205615373227, 5.723579033469881, 6.159848659322155, 6.811448049397319, 7.252853236762221, 7.969246411007359, 8.380510619368602, 9.052815520209135, 9.672195924872420, 10.13292876349413, 10.64725377836869, 11.18858135017921, 11.69999347601947, 12.11154268153246, 12.68603730360306, 13.07876346249368, 13.76610331134312, 14.34716766827213, 14.57979846338424
