L(s) = 1 | + 7-s − 6·11-s − 13-s + 3·17-s − 4·19-s + 3·23-s + 3·29-s − 5·31-s − 10·37-s − 9·41-s + 43-s + 49-s − 9·53-s − 9·59-s − 11·61-s + 4·67-s − 12·71-s + 10·73-s − 6·77-s + 10·79-s + 9·83-s + 6·89-s − 91-s − 14·97-s + 101-s + 103-s + 107-s + ⋯ |
L(s) = 1 | + 0.377·7-s − 1.80·11-s − 0.277·13-s + 0.727·17-s − 0.917·19-s + 0.625·23-s + 0.557·29-s − 0.898·31-s − 1.64·37-s − 1.40·41-s + 0.152·43-s + 1/7·49-s − 1.23·53-s − 1.17·59-s − 1.40·61-s + 0.488·67-s − 1.42·71-s + 1.17·73-s − 0.683·77-s + 1.12·79-s + 0.987·83-s + 0.635·89-s − 0.104·91-s − 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6019414394\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6019414394\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 + 11 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−13.77447145511060, −13.25131504462570, −12.73528927964535, −12.30299074168303, −11.98274130194934, −11.01649405518251, −10.87103120325348, −10.34724775935695, −9.973921092616036, −9.240553416773350, −8.714450138068026, −8.229810369342053, −7.691138493495677, −7.407169714452295, −6.666628217026224, −6.128948585077189, −5.427852018741320, −4.902261956407778, −4.809999229602979, −3.735573219679289, −3.230275290884830, −2.633983432167070, −1.978243794037038, −1.366754213165334, −0.2351288145553258,
0.2351288145553258, 1.366754213165334, 1.978243794037038, 2.633983432167070, 3.230275290884830, 3.735573219679289, 4.809999229602979, 4.902261956407778, 5.427852018741320, 6.128948585077189, 6.666628217026224, 7.407169714452295, 7.691138493495677, 8.229810369342053, 8.714450138068026, 9.240553416773350, 9.973921092616036, 10.34724775935695, 10.87103120325348, 11.01649405518251, 11.98274130194934, 12.30299074168303, 12.73528927964535, 13.25131504462570, 13.77447145511060