| L(s) = 1 | − 3·3-s − 5-s + 6·9-s + 5·11-s + 3·15-s + 17-s + 6·19-s + 6·23-s + 25-s − 9·27-s − 9·29-s − 4·31-s − 15·33-s − 2·37-s − 4·41-s + 10·43-s − 6·45-s − 47-s − 3·51-s + 4·53-s − 5·55-s − 18·57-s − 8·59-s + 8·61-s − 12·67-s − 18·69-s − 8·71-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 0.447·5-s + 2·9-s + 1.50·11-s + 0.774·15-s + 0.242·17-s + 1.37·19-s + 1.25·23-s + 1/5·25-s − 1.73·27-s − 1.67·29-s − 0.718·31-s − 2.61·33-s − 0.328·37-s − 0.624·41-s + 1.52·43-s − 0.894·45-s − 0.145·47-s − 0.420·51-s + 0.549·53-s − 0.674·55-s − 2.38·57-s − 1.04·59-s + 1.02·61-s − 1.46·67-s − 2.16·69-s − 0.949·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 165620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + p T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 13 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
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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
−13.41176157867525, −12.79435371017204, −12.32219462726335, −11.98910301526820, −11.62423156111360, −11.07747871127964, −10.95428132306029, −10.32514646137705, −9.605839112349209, −9.291076255779090, −8.925596108564376, −8.012811083572068, −7.426223285855941, −6.976278836333336, −6.823107902526324, −5.945881294604564, −5.671025549855443, −5.208262608754405, −4.601015035636601, −4.069925424260568, −3.580853299990881, −2.989128291921713, −1.843037135396372, −1.268830784358413, −0.8184066753420014, 0,
0.8184066753420014, 1.268830784358413, 1.843037135396372, 2.989128291921713, 3.580853299990881, 4.069925424260568, 4.601015035636601, 5.208262608754405, 5.671025549855443, 5.945881294604564, 6.823107902526324, 6.976278836333336, 7.426223285855941, 8.012811083572068, 8.925596108564376, 9.291076255779090, 9.605839112349209, 10.32514646137705, 10.95428132306029, 11.07747871127964, 11.62423156111360, 11.98910301526820, 12.32219462726335, 12.79435371017204, 13.41176157867525
