| L(s) = 1 | − 0.218·2-s − 1.95·4-s − 4.66·7-s + 0.865·8-s + 2.85·11-s − 1.26·13-s + 1.02·14-s + 3.71·16-s + 4.33·17-s − 4.96·19-s − 0.624·22-s + 0.202·23-s + 0.276·26-s + 9.11·28-s − 4.57·29-s − 31-s − 2.54·32-s − 0.948·34-s + 2.16·37-s + 1.08·38-s − 10.0·41-s + 9.47·43-s − 5.56·44-s − 0.0442·46-s + 7.86·47-s + 14.7·49-s + 2.46·52-s + ⋯ |
| L(s) = 1 | − 0.154·2-s − 0.976·4-s − 1.76·7-s + 0.305·8-s + 0.860·11-s − 0.350·13-s + 0.273·14-s + 0.928·16-s + 1.05·17-s − 1.13·19-s − 0.133·22-s + 0.0421·23-s + 0.0543·26-s + 1.72·28-s − 0.849·29-s − 0.179·31-s − 0.449·32-s − 0.162·34-s + 0.355·37-s + 0.176·38-s − 1.56·41-s + 1.44·43-s − 0.839·44-s − 0.00652·46-s + 1.14·47-s + 2.11·49-s + 0.342·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 31 | \( 1 + T \) |
| good | 2 | \( 1 + 0.218T + 2T^{2} \) |
| 7 | \( 1 + 4.66T + 7T^{2} \) |
| 11 | \( 1 - 2.85T + 11T^{2} \) |
| 13 | \( 1 + 1.26T + 13T^{2} \) |
| 17 | \( 1 - 4.33T + 17T^{2} \) |
| 19 | \( 1 + 4.96T + 19T^{2} \) |
| 23 | \( 1 - 0.202T + 23T^{2} \) |
| 29 | \( 1 + 4.57T + 29T^{2} \) |
| 37 | \( 1 - 2.16T + 37T^{2} \) |
| 41 | \( 1 + 10.0T + 41T^{2} \) |
| 43 | \( 1 - 9.47T + 43T^{2} \) |
| 47 | \( 1 - 7.86T + 47T^{2} \) |
| 53 | \( 1 - 9.93T + 53T^{2} \) |
| 59 | \( 1 - 5.58T + 59T^{2} \) |
| 61 | \( 1 + 9.27T + 61T^{2} \) |
| 67 | \( 1 - 2.37T + 67T^{2} \) |
| 71 | \( 1 - 7.06T + 71T^{2} \) |
| 73 | \( 1 + 6.73T + 73T^{2} \) |
| 79 | \( 1 - 0.948T + 79T^{2} \) |
| 83 | \( 1 + 6.68T + 83T^{2} \) |
| 89 | \( 1 + 3.42T + 89T^{2} \) |
| 97 | \( 1 - 14.0T + 97T^{2} \) |
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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
−7.51063058526250359322680312065, −6.97267929351284471569745459660, −6.07120368144239196209784717343, −5.68671215329492928350541204094, −4.62543604740471691703452268374, −3.79753650535797788600876116964, −3.45112061559132309247663750306, −2.34462586606114810103222001914, −0.983674992687824988740890669320, 0,
0.983674992687824988740890669320, 2.34462586606114810103222001914, 3.45112061559132309247663750306, 3.79753650535797788600876116964, 4.62543604740471691703452268374, 5.68671215329492928350541204094, 6.07120368144239196209784717343, 6.97267929351284471569745459660, 7.51063058526250359322680312065
