L(s) = 1 | − 3-s − 7-s + 9-s + 5·11-s − 2·13-s + 2·17-s − 19-s + 21-s + 23-s − 27-s − 4·29-s − 5·33-s + 6·37-s + 2·39-s − 3·41-s + 2·43-s + 13·47-s + 49-s − 2·51-s + 3·53-s + 57-s − 4·59-s − 10·61-s − 63-s − 14·67-s − 69-s + 14·71-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.377·7-s + 1/3·9-s + 1.50·11-s − 0.554·13-s + 0.485·17-s − 0.229·19-s + 0.218·21-s + 0.208·23-s − 0.192·27-s − 0.742·29-s − 0.870·33-s + 0.986·37-s + 0.320·39-s − 0.468·41-s + 0.304·43-s + 1.89·47-s + 1/7·49-s − 0.280·51-s + 0.412·53-s + 0.132·57-s − 0.520·59-s − 1.28·61-s − 0.125·63-s − 1.71·67-s − 0.120·69-s + 1.66·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 96600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 96600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.907880741\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.907880741\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
good | 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 13 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 - 14 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 17 T + p T^{2} \) |
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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
−13.84909707419028, −13.32073980994898, −12.56230279036080, −12.28835327672701, −12.01032964398618, −11.20646716033969, −11.04025592774501, −10.34734115110323, −9.731557345021174, −9.427258036343824, −8.954560749285970, −8.334841310853783, −7.567481623784101, −7.213234246048412, −6.661358980799688, −6.116127701922980, −5.727372360077548, −5.088845068911674, −4.335992200628220, −4.032066972138473, −3.343502613153122, −2.650504156156603, −1.862549858631614, −1.179464816356095, −0.4980866842034270,
0.4980866842034270, 1.179464816356095, 1.862549858631614, 2.650504156156603, 3.343502613153122, 4.032066972138473, 4.335992200628220, 5.088845068911674, 5.727372360077548, 6.116127701922980, 6.661358980799688, 7.213234246048412, 7.567481623784101, 8.334841310853783, 8.954560749285970, 9.427258036343824, 9.731557345021174, 10.34734115110323, 11.04025592774501, 11.20646716033969, 12.01032964398618, 12.28835327672701, 12.56230279036080, 13.32073980994898, 13.84909707419028