| L(s) = 1 | − 4·5-s − 49·7-s − 370·11-s + 122·13-s + 1.42e3·17-s + 1.72e3·19-s + 2.67e3·23-s − 3.10e3·25-s − 4.30e3·29-s + 3.10e3·31-s + 196·35-s − 1.43e4·37-s + 1.22e4·41-s − 2.16e4·43-s + 2.64e3·47-s + 2.40e3·49-s + 2.43e4·53-s + 1.48e3·55-s + 1.40e4·59-s − 2.44e4·61-s − 488·65-s + 7.20e3·67-s − 5.43e4·71-s − 4.89e4·73-s + 1.81e4·77-s − 3.33e4·79-s + 4.00e3·83-s + ⋯ |
| L(s) = 1 | − 0.0715·5-s − 0.377·7-s − 0.921·11-s + 0.200·13-s + 1.19·17-s + 1.09·19-s + 1.05·23-s − 0.994·25-s − 0.949·29-s + 0.580·31-s + 0.0270·35-s − 1.71·37-s + 1.14·41-s − 1.78·43-s + 0.174·47-s + 1/7·49-s + 1.19·53-s + 0.0659·55-s + 0.526·59-s − 0.842·61-s − 0.0143·65-s + 0.196·67-s − 1.27·71-s − 1.07·73-s + 0.348·77-s − 0.600·79-s + 0.0637·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + p^{2} T \) |
| good | 5 | \( 1 + 4 T + p^{5} T^{2} \) |
| 11 | \( 1 + 370 T + p^{5} T^{2} \) |
| 13 | \( 1 - 122 T + p^{5} T^{2} \) |
| 17 | \( 1 - 84 p T + p^{5} T^{2} \) |
| 19 | \( 1 - 1724 T + p^{5} T^{2} \) |
| 23 | \( 1 - 2670 T + p^{5} T^{2} \) |
| 29 | \( 1 + 4302 T + p^{5} T^{2} \) |
| 31 | \( 1 - 3104 T + p^{5} T^{2} \) |
| 37 | \( 1 + 14318 T + p^{5} T^{2} \) |
| 41 | \( 1 - 12272 T + p^{5} T^{2} \) |
| 43 | \( 1 + 21652 T + p^{5} T^{2} \) |
| 47 | \( 1 - 2644 T + p^{5} T^{2} \) |
| 53 | \( 1 - 24342 T + p^{5} T^{2} \) |
| 59 | \( 1 - 14088 T + p^{5} T^{2} \) |
| 61 | \( 1 + 24474 T + p^{5} T^{2} \) |
| 67 | \( 1 - 7208 T + p^{5} T^{2} \) |
| 71 | \( 1 + 54302 T + p^{5} T^{2} \) |
| 73 | \( 1 + 48962 T + p^{5} T^{2} \) |
| 79 | \( 1 + 33332 T + p^{5} T^{2} \) |
| 83 | \( 1 - 4004 T + p^{5} T^{2} \) |
| 89 | \( 1 + 64752 T + p^{5} T^{2} \) |
| 97 | \( 1 - 7038 T + p^{5} T^{2} \) |
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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
−9.808270293239071817311274902090, −8.804647581131339940212341835351, −7.77087035347869781574366247572, −7.10424151740721878739699333015, −5.78299212788107048199640951638, −5.12744999290820249785752853904, −3.68343188644875717717062835541, −2.82213678112788875559805119318, −1.33869515484221135336126098712, 0,
1.33869515484221135336126098712, 2.82213678112788875559805119318, 3.68343188644875717717062835541, 5.12744999290820249785752853904, 5.78299212788107048199640951638, 7.10424151740721878739699333015, 7.77087035347869781574366247572, 8.804647581131339940212341835351, 9.808270293239071817311274902090
