| L(s) = 1 | − 3-s + 2·7-s + 9-s + 2·11-s + 5.23·13-s + 0.381·17-s + 3.85·19-s − 2·21-s + 7.61·23-s − 27-s + 2.47·29-s − 8.09·31-s − 2·33-s − 8.94·37-s − 5.23·39-s − 9.70·41-s − 5.23·43-s + 8.61·47-s − 3·49-s − 0.381·51-s + 10.3·53-s − 3.85·57-s + 4·59-s − 2.14·61-s + 2·63-s + 2.47·67-s − 7.61·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.755·7-s + 0.333·9-s + 0.603·11-s + 1.45·13-s + 0.0926·17-s + 0.884·19-s − 0.436·21-s + 1.58·23-s − 0.192·27-s + 0.459·29-s − 1.45·31-s − 0.348·33-s − 1.47·37-s − 0.838·39-s − 1.51·41-s − 0.798·43-s + 1.25·47-s − 0.428·49-s − 0.0534·51-s + 1.41·53-s − 0.510·57-s + 0.520·59-s − 0.274·61-s + 0.251·63-s + 0.302·67-s − 0.917·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.983783777\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.983783777\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 7 | \( 1 - 2T + 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 - 5.23T + 13T^{2} \) |
| 17 | \( 1 - 0.381T + 17T^{2} \) |
| 19 | \( 1 - 3.85T + 19T^{2} \) |
| 23 | \( 1 - 7.61T + 23T^{2} \) |
| 29 | \( 1 - 2.47T + 29T^{2} \) |
| 31 | \( 1 + 8.09T + 31T^{2} \) |
| 37 | \( 1 + 8.94T + 37T^{2} \) |
| 41 | \( 1 + 9.70T + 41T^{2} \) |
| 43 | \( 1 + 5.23T + 43T^{2} \) |
| 47 | \( 1 - 8.61T + 47T^{2} \) |
| 53 | \( 1 - 10.3T + 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 + 2.14T + 61T^{2} \) |
| 67 | \( 1 - 2.47T + 67T^{2} \) |
| 71 | \( 1 - 7.70T + 71T^{2} \) |
| 73 | \( 1 - 7.70T + 73T^{2} \) |
| 79 | \( 1 + 1.09T + 79T^{2} \) |
| 83 | \( 1 - 4.61T + 83T^{2} \) |
| 89 | \( 1 - 7.70T + 89T^{2} \) |
| 97 | \( 1 - 4.76T + 97T^{2} \) |
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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
−8.781182970935087876160193834153, −8.031236625730183765512644093588, −7.02676399489855611427944843089, −6.60309407657920898869297798758, −5.42809963085353009721213431589, −5.15235391209609133221610787549, −3.95799879473017197447544644161, −3.29966310376399475953764791190, −1.75774167150624577489283223705, −0.974661879025616798555728787660,
0.974661879025616798555728787660, 1.75774167150624577489283223705, 3.29966310376399475953764791190, 3.95799879473017197447544644161, 5.15235391209609133221610787549, 5.42809963085353009721213431589, 6.60309407657920898869297798758, 7.02676399489855611427944843089, 8.031236625730183765512644093588, 8.781182970935087876160193834153
