| L(s) = 1 | − 211·7-s − 427·13-s + 3.14e3·19-s + 7.60e3·31-s + 1.65e4·37-s − 1.91e4·43-s + 2.77e4·49-s − 1.83e4·61-s − 3.79e4·67-s − 1.45e3·73-s − 1.00e5·79-s + 9.00e4·91-s − 4.33e4·97-s + 1.40e5·103-s − 2.47e5·109-s + ⋯ |
| L(s) = 1 | − 1.62·7-s − 0.700·13-s + 1.99·19-s + 1.42·31-s + 1.98·37-s − 1.57·43-s + 1.64·49-s − 0.629·61-s − 1.03·67-s − 0.0318·73-s − 1.81·79-s + 1.14·91-s − 0.467·97-s + 1.30·103-s − 1.99·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 211 T + p^{5} T^{2} \) |
| 11 | \( 1 + p^{5} T^{2} \) |
| 13 | \( 1 + 427 T + p^{5} T^{2} \) |
| 17 | \( 1 + p^{5} T^{2} \) |
| 19 | \( 1 - 3143 T + p^{5} T^{2} \) |
| 23 | \( 1 + p^{5} T^{2} \) |
| 29 | \( 1 + p^{5} T^{2} \) |
| 31 | \( 1 - 7601 T + p^{5} T^{2} \) |
| 37 | \( 1 - 16550 T + p^{5} T^{2} \) |
| 41 | \( 1 + p^{5} T^{2} \) |
| 43 | \( 1 + 19123 T + p^{5} T^{2} \) |
| 47 | \( 1 + p^{5} T^{2} \) |
| 53 | \( 1 + p^{5} T^{2} \) |
| 59 | \( 1 + p^{5} T^{2} \) |
| 61 | \( 1 + 18301 T + p^{5} T^{2} \) |
| 67 | \( 1 + 37939 T + p^{5} T^{2} \) |
| 71 | \( 1 + p^{5} T^{2} \) |
| 73 | \( 1 + 1450 T + p^{5} T^{2} \) |
| 79 | \( 1 + 100564 T + p^{5} T^{2} \) |
| 83 | \( 1 + p^{5} T^{2} \) |
| 89 | \( 1 + p^{5} T^{2} \) |
| 97 | \( 1 + 43339 T + p^{5} T^{2} \) |
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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
−9.248055821497047211596274382212, −8.012065665931652031459545010393, −7.17780549071884043582082042315, −6.40123337859740177734187699881, −5.55802303505846083787717947544, −4.47064519729459710839186531965, −3.25718799000678328329880170366, −2.71645172039119248264447993463, −1.07902257137737312906068135223, 0,
1.07902257137737312906068135223, 2.71645172039119248264447993463, 3.25718799000678328329880170366, 4.47064519729459710839186531965, 5.55802303505846083787717947544, 6.40123337859740177734187699881, 7.17780549071884043582082042315, 8.012065665931652031459545010393, 9.248055821497047211596274382212
