| L(s) = 1 | + 25·5-s − 28·7-s + 208·11-s − 422·13-s + 146·17-s − 2.01e3·19-s + 1.09e3·23-s + 625·25-s + 1.46e3·29-s − 80·31-s − 700·35-s − 1.57e4·37-s + 2.35e3·41-s + 2.81e3·43-s + 7.96e3·47-s − 1.60e4·49-s + 7.59e3·53-s + 5.20e3·55-s − 1.80e4·59-s − 1.96e4·61-s − 1.05e4·65-s + 3.18e4·67-s − 5.72e4·71-s + 9.90e3·73-s − 5.82e3·77-s + 7.87e3·79-s − 1.09e5·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.215·7-s + 0.518·11-s − 0.692·13-s + 0.122·17-s − 1.27·19-s + 0.432·23-s + 1/5·25-s + 0.322·29-s − 0.0149·31-s − 0.0965·35-s − 1.89·37-s + 0.219·41-s + 0.231·43-s + 0.525·47-s − 0.953·49-s + 0.371·53-s + 0.231·55-s − 0.675·59-s − 0.676·61-s − 0.309·65-s + 0.867·67-s − 1.34·71-s + 0.217·73-s − 0.111·77-s + 0.141·79-s − 1.75·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - p^{2} T \) |
| good | 7 | \( 1 + 4 p T + p^{5} T^{2} \) |
| 11 | \( 1 - 208 T + p^{5} T^{2} \) |
| 13 | \( 1 + 422 T + p^{5} T^{2} \) |
| 17 | \( 1 - 146 T + p^{5} T^{2} \) |
| 19 | \( 1 + 2012 T + p^{5} T^{2} \) |
| 23 | \( 1 - 1096 T + p^{5} T^{2} \) |
| 29 | \( 1 - 1462 T + p^{5} T^{2} \) |
| 31 | \( 1 + 80 T + p^{5} T^{2} \) |
| 37 | \( 1 + 15750 T + p^{5} T^{2} \) |
| 41 | \( 1 - 2358 T + p^{5} T^{2} \) |
| 43 | \( 1 - 2812 T + p^{5} T^{2} \) |
| 47 | \( 1 - 7960 T + p^{5} T^{2} \) |
| 53 | \( 1 - 7590 T + p^{5} T^{2} \) |
| 59 | \( 1 + 18064 T + p^{5} T^{2} \) |
| 61 | \( 1 + 19658 T + p^{5} T^{2} \) |
| 67 | \( 1 - 31868 T + p^{5} T^{2} \) |
| 71 | \( 1 + 57216 T + p^{5} T^{2} \) |
| 73 | \( 1 - 9906 T + p^{5} T^{2} \) |
| 79 | \( 1 - 7872 T + p^{5} T^{2} \) |
| 83 | \( 1 + 109996 T + p^{5} T^{2} \) |
| 89 | \( 1 + 62466 T + p^{5} T^{2} \) |
| 97 | \( 1 + 97598 T + p^{5} T^{2} \) |
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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
−10.15734796753416966552085396520, −9.250421795947785558122726345588, −8.422691168411238480185324801269, −7.15677389166126570550238019442, −6.36103366823240715579148360433, −5.24661060442133768835711735748, −4.12764863371030155836086371653, −2.79045226164779563708159656080, −1.57128527062264469790520847087, 0,
1.57128527062264469790520847087, 2.79045226164779563708159656080, 4.12764863371030155836086371653, 5.24661060442133768835711735748, 6.36103366823240715579148360433, 7.15677389166126570550238019442, 8.422691168411238480185324801269, 9.250421795947785558122726345588, 10.15734796753416966552085396520
