| L(s) = 1 | + 2·7-s + 11-s + 4·17-s + 4·19-s − 4·23-s − 6·29-s + 2·37-s − 2·41-s − 2·43-s − 12·47-s − 3·49-s − 2·53-s + 4·59-s + 2·61-s + 8·71-s + 2·77-s − 8·79-s − 2·83-s + 2·89-s + 2·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + 8·119-s + ⋯ |
| L(s) = 1 | + 0.755·7-s + 0.301·11-s + 0.970·17-s + 0.917·19-s − 0.834·23-s − 1.11·29-s + 0.328·37-s − 0.312·41-s − 0.304·43-s − 1.75·47-s − 3/7·49-s − 0.274·53-s + 0.520·59-s + 0.256·61-s + 0.949·71-s + 0.227·77-s − 0.900·79-s − 0.219·83-s + 0.211·89-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + 0.733·119-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 39600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−14.75891467611760, −14.65922267213952, −14.11911234413241, −13.53870993978922, −13.01229217261126, −12.39372094294056, −11.84559272540542, −11.41376526024748, −11.05482693615774, −10.17249476776229, −9.827434864204804, −9.325918756771155, −8.584436907110632, −7.999573953769081, −7.697838545984963, −7.022334914890664, −6.349066239389953, −5.700764676724669, −5.172966811218124, −4.660252618775051, −3.771328831684229, −3.409245911052587, −2.500091344761351, −1.674903355134697, −1.167743675350700, 0,
1.167743675350700, 1.674903355134697, 2.500091344761351, 3.409245911052587, 3.771328831684229, 4.660252618775051, 5.172966811218124, 5.700764676724669, 6.349066239389953, 7.022334914890664, 7.697838545984963, 7.999573953769081, 8.584436907110632, 9.325918756771155, 9.827434864204804, 10.17249476776229, 11.05482693615774, 11.41376526024748, 11.84559272540542, 12.39372094294056, 13.01229217261126, 13.53870993978922, 14.11911234413241, 14.65922267213952, 14.75891467611760
