L(s) = 1 | + 3-s − 4·7-s + 9-s − 2·13-s − 6·17-s + 4·19-s − 4·21-s + 27-s − 6·29-s − 8·31-s − 2·37-s − 2·39-s − 6·41-s − 4·43-s + 9·49-s − 6·51-s + 6·53-s + 4·57-s − 10·61-s − 4·63-s − 4·67-s − 2·73-s − 8·79-s + 81-s + 12·83-s − 6·87-s + 18·89-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 1.51·7-s + 1/3·9-s − 0.554·13-s − 1.45·17-s + 0.917·19-s − 0.872·21-s + 0.192·27-s − 1.11·29-s − 1.43·31-s − 0.328·37-s − 0.320·39-s − 0.937·41-s − 0.609·43-s + 9/7·49-s − 0.840·51-s + 0.824·53-s + 0.529·57-s − 1.28·61-s − 0.503·63-s − 0.488·67-s − 0.234·73-s − 0.900·79-s + 1/9·81-s + 1.31·83-s − 0.643·87-s + 1.90·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−9.276841907117526756164869085518, −8.811879039450229959581280002555, −7.53023481714424969283045906095, −6.97121259233254998318201926571, −6.10897773314734163440451778623, −5.03186235211519720784151262088, −3.82839635062702052518767893213, −3.10714236804866821433728477742, −2.01462355044529552405323097963, 0,
2.01462355044529552405323097963, 3.10714236804866821433728477742, 3.82839635062702052518767893213, 5.03186235211519720784151262088, 6.10897773314734163440451778623, 6.97121259233254998318201926571, 7.53023481714424969283045906095, 8.811879039450229959581280002555, 9.276841907117526756164869085518