L(s) = 1 | − 3-s − 2·9-s − 6·11-s + 2·13-s − 6·17-s + 8·19-s + 3·23-s + 5·27-s + 3·29-s + 2·31-s + 6·33-s − 8·37-s − 2·39-s + 3·41-s + 5·43-s + 6·51-s − 12·53-s − 8·57-s + 61-s − 7·67-s − 3·69-s − 10·73-s + 4·79-s + 81-s − 3·83-s − 3·87-s + 3·89-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 2/3·9-s − 1.80·11-s + 0.554·13-s − 1.45·17-s + 1.83·19-s + 0.625·23-s + 0.962·27-s + 0.557·29-s + 0.359·31-s + 1.04·33-s − 1.31·37-s − 0.320·39-s + 0.468·41-s + 0.762·43-s + 0.840·51-s − 1.64·53-s − 1.05·57-s + 0.128·61-s − 0.855·67-s − 0.361·69-s − 1.17·73-s + 0.450·79-s + 1/9·81-s − 0.329·83-s − 0.321·87-s + 0.317·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−15.96793841437959, −15.64690026263608, −15.02650659365112, −14.13988592277478, −13.73518992646548, −13.30528600227716, −12.63864142379840, −12.13668571681092, −11.30945791957657, −11.13747911755331, −10.52072606649344, −9.997974295336216, −9.156678522446620, −8.675645425977525, −8.045416645060497, −7.425536746940333, −6.832536413100543, −6.029938520981934, −5.567650188804533, −4.945639462656363, −4.504291378310743, −3.212009778679754, −2.940449792485784, −2.028870947452467, −0.8787372296875277, 0,
0.8787372296875277, 2.028870947452467, 2.940449792485784, 3.212009778679754, 4.504291378310743, 4.945639462656363, 5.567650188804533, 6.029938520981934, 6.832536413100543, 7.425536746940333, 8.045416645060497, 8.675645425977525, 9.156678522446620, 9.997974295336216, 10.52072606649344, 11.13747911755331, 11.30945791957657, 12.13668571681092, 12.63864142379840, 13.30528600227716, 13.73518992646548, 14.13988592277478, 15.02650659365112, 15.64690026263608, 15.96793841437959