L(s) = 1 | + 5-s + 4·7-s − 11-s − 4·13-s + 6·17-s − 2·19-s + 25-s + 4·31-s + 4·35-s − 10·37-s + 4·43-s + 12·47-s + 9·49-s − 6·53-s − 55-s + 12·59-s − 10·61-s − 4·65-s + 4·67-s + 8·73-s − 4·77-s + 10·79-s − 6·83-s + 6·85-s + 6·89-s − 16·91-s − 2·95-s + ⋯ |
L(s) = 1 | + 0.447·5-s + 1.51·7-s − 0.301·11-s − 1.10·13-s + 1.45·17-s − 0.458·19-s + 1/5·25-s + 0.718·31-s + 0.676·35-s − 1.64·37-s + 0.609·43-s + 1.75·47-s + 9/7·49-s − 0.824·53-s − 0.134·55-s + 1.56·59-s − 1.28·61-s − 0.496·65-s + 0.488·67-s + 0.936·73-s − 0.455·77-s + 1.12·79-s − 0.658·83-s + 0.650·85-s + 0.635·89-s − 1.67·91-s − 0.205·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.728735533\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.728735533\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 + T \) |
good | 7 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−7.63128159984411859766401927238, −7.49188750183322789640808510914, −6.43396394164385775632370324189, −5.54463651750505688212489619037, −5.09825101818598663261124464215, −4.50687286458168761339703509509, −3.50696644772247377735915274092, −2.48094670311445046493521919375, −1.84571580333824316491649626033, −0.844318620736814678494553967354,
0.844318620736814678494553967354, 1.84571580333824316491649626033, 2.48094670311445046493521919375, 3.50696644772247377735915274092, 4.50687286458168761339703509509, 5.09825101818598663261124464215, 5.54463651750505688212489619037, 6.43396394164385775632370324189, 7.49188750183322789640808510914, 7.63128159984411859766401927238