| L(s) = 1 | + 1.73·5-s − 7-s + 1.73·11-s + 2·13-s − 6.92·17-s − 5·19-s − 1.73·23-s − 2.00·25-s − 10.3·29-s − 5·31-s − 1.73·35-s − 7·37-s + 5.19·41-s + 4·43-s − 6.92·47-s + 49-s + 13.8·53-s + 2.99·55-s + 6.92·59-s + 8·61-s + 3.46·65-s − 14·67-s − 5.19·71-s − 4·73-s − 1.73·77-s − 8·79-s + 10.3·83-s + ⋯ |
| L(s) = 1 | + 0.774·5-s − 0.377·7-s + 0.522·11-s + 0.554·13-s − 1.68·17-s − 1.14·19-s − 0.361·23-s − 0.400·25-s − 1.92·29-s − 0.898·31-s − 0.292·35-s − 1.15·37-s + 0.811·41-s + 0.609·43-s − 1.01·47-s + 0.142·49-s + 1.90·53-s + 0.404·55-s + 0.901·59-s + 1.02·61-s + 0.429·65-s − 1.71·67-s − 0.616·71-s − 0.468·73-s − 0.197·77-s − 0.900·79-s + 1.14·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 5 | \( 1 - 1.73T + 5T^{2} \) |
| 11 | \( 1 - 1.73T + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 + 6.92T + 17T^{2} \) |
| 19 | \( 1 + 5T + 19T^{2} \) |
| 23 | \( 1 + 1.73T + 23T^{2} \) |
| 29 | \( 1 + 10.3T + 29T^{2} \) |
| 31 | \( 1 + 5T + 31T^{2} \) |
| 37 | \( 1 + 7T + 37T^{2} \) |
| 41 | \( 1 - 5.19T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 + 6.92T + 47T^{2} \) |
| 53 | \( 1 - 13.8T + 53T^{2} \) |
| 59 | \( 1 - 6.92T + 59T^{2} \) |
| 61 | \( 1 - 8T + 61T^{2} \) |
| 67 | \( 1 + 14T + 67T^{2} \) |
| 71 | \( 1 + 5.19T + 71T^{2} \) |
| 73 | \( 1 + 4T + 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 - 10.3T + 83T^{2} \) |
| 89 | \( 1 + 8.66T + 89T^{2} \) |
| 97 | \( 1 + 4T + 97T^{2} \) |
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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
−8.734309531278504364446029260553, −7.46007646043881215382457330062, −6.74559018645882667637565563622, −6.06818255571352903426599549803, −5.48320358827353498153050842212, −4.25443295202664137809222498346, −3.72295817293409708713900409295, −2.34646502498543617185039823869, −1.73296615714503807277498229593, 0,
1.73296615714503807277498229593, 2.34646502498543617185039823869, 3.72295817293409708713900409295, 4.25443295202664137809222498346, 5.48320358827353498153050842212, 6.06818255571352903426599549803, 6.74559018645882667637565563622, 7.46007646043881215382457330062, 8.734309531278504364446029260553
