L(s) = 1 | + 3-s − 5-s + 3·7-s − 2·9-s − 11-s + 6·13-s − 15-s − 7·17-s − 5·19-s + 3·21-s − 6·23-s + 25-s − 5·27-s − 5·29-s − 3·31-s − 33-s − 3·35-s − 3·37-s + 6·39-s + 2·41-s − 4·43-s + 2·45-s − 2·47-s + 2·49-s − 7·51-s + 53-s + 55-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1.13·7-s − 2/3·9-s − 0.301·11-s + 1.66·13-s − 0.258·15-s − 1.69·17-s − 1.14·19-s + 0.654·21-s − 1.25·23-s + 1/5·25-s − 0.962·27-s − 0.928·29-s − 0.538·31-s − 0.174·33-s − 0.507·35-s − 0.493·37-s + 0.960·39-s + 0.312·41-s − 0.609·43-s + 0.298·45-s − 0.291·47-s + 2/7·49-s − 0.980·51-s + 0.137·53-s + 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 + T \) |
good | 3 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 7 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−8.357814177462641292320769382096, −7.77905152525463235597143862770, −6.69775542583099421630453754851, −5.99960131598184263414621197219, −5.10579515322739311989630485935, −4.12435800827008436491655987371, −3.66964935092825906643887488230, −2.37223324814799152265566245541, −1.71535809302257164290595253608, 0,
1.71535809302257164290595253608, 2.37223324814799152265566245541, 3.66964935092825906643887488230, 4.12435800827008436491655987371, 5.10579515322739311989630485935, 5.99960131598184263414621197219, 6.69775542583099421630453754851, 7.77905152525463235597143862770, 8.357814177462641292320769382096