| L(s) = 1 | + 34·5-s − 49·7-s + 332·11-s − 1.02e3·13-s − 922·17-s + 452·19-s + 3.77e3·23-s − 1.96e3·25-s − 1.16e3·29-s − 9.79e3·31-s − 1.66e3·35-s + 2.39e3·37-s + 7.23e3·41-s + 4.65e3·43-s − 2.46e4·47-s + 2.40e3·49-s − 1.11e3·53-s + 1.12e4·55-s − 4.68e4·59-s − 9.76e3·61-s − 3.48e4·65-s − 2.62e4·67-s − 6.54e4·71-s − 5.60e3·73-s − 1.62e4·77-s − 9.84e3·79-s − 6.11e4·83-s + ⋯ |
| L(s) = 1 | + 0.608·5-s − 0.377·7-s + 0.827·11-s − 1.68·13-s − 0.773·17-s + 0.287·19-s + 1.48·23-s − 0.630·25-s − 0.257·29-s − 1.83·31-s − 0.229·35-s + 0.287·37-s + 0.671·41-s + 0.383·43-s − 1.62·47-s + 1/7·49-s − 0.0542·53-s + 0.503·55-s − 1.75·59-s − 0.335·61-s − 1.02·65-s − 0.714·67-s − 1.54·71-s − 0.123·73-s − 0.312·77-s − 0.177·79-s − 0.973·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 34 T + p^{5} T^{2} \) |
| 11 | \( 1 - 332 T + p^{5} T^{2} \) |
| 13 | \( 1 + 1026 T + p^{5} T^{2} \) |
| 17 | \( 1 + 922 T + p^{5} T^{2} \) |
| 19 | \( 1 - 452 T + p^{5} T^{2} \) |
| 23 | \( 1 - 3776 T + p^{5} T^{2} \) |
| 29 | \( 1 + 1166 T + p^{5} T^{2} \) |
| 31 | \( 1 + 9792 T + p^{5} T^{2} \) |
| 37 | \( 1 - 2390 T + p^{5} T^{2} \) |
| 41 | \( 1 - 7230 T + p^{5} T^{2} \) |
| 43 | \( 1 - 4652 T + p^{5} T^{2} \) |
| 47 | \( 1 + 24672 T + p^{5} T^{2} \) |
| 53 | \( 1 + 1110 T + p^{5} T^{2} \) |
| 59 | \( 1 + 46892 T + p^{5} T^{2} \) |
| 61 | \( 1 + 9762 T + p^{5} T^{2} \) |
| 67 | \( 1 + 26252 T + p^{5} T^{2} \) |
| 71 | \( 1 + 65440 T + p^{5} T^{2} \) |
| 73 | \( 1 + 5606 T + p^{5} T^{2} \) |
| 79 | \( 1 + 9840 T + p^{5} T^{2} \) |
| 83 | \( 1 + 61108 T + p^{5} T^{2} \) |
| 89 | \( 1 - 62958 T + p^{5} T^{2} \) |
| 97 | \( 1 + 37838 T + p^{5} T^{2} \) |
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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
−10.67801095880390968041019569664, −9.471608992046821572195457984735, −9.191505614547349917516899781348, −7.55006272164724197498993845134, −6.73298255321240528839049784307, −5.57970983211582409977470066376, −4.46292149411776776270220221384, −2.97761298034663055434348945471, −1.72011968859156978245597450097, 0,
1.72011968859156978245597450097, 2.97761298034663055434348945471, 4.46292149411776776270220221384, 5.57970983211582409977470066376, 6.73298255321240528839049784307, 7.55006272164724197498993845134, 9.191505614547349917516899781348, 9.471608992046821572195457984735, 10.67801095880390968041019569664
