| L(s) = 1 | + 7-s − 3·11-s + 13-s − 5·17-s + 2·19-s + 3·23-s + 4·31-s − 37-s − 9·41-s + 2·43-s − 8·47-s − 6·49-s − 53-s + 4·59-s − 3·61-s − 16·67-s − 15·71-s + 14·73-s − 3·77-s + 15·79-s + 8·83-s + 9·89-s + 91-s + 5·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | + 0.377·7-s − 0.904·11-s + 0.277·13-s − 1.21·17-s + 0.458·19-s + 0.625·23-s + 0.718·31-s − 0.164·37-s − 1.40·41-s + 0.304·43-s − 1.16·47-s − 6/7·49-s − 0.137·53-s + 0.520·59-s − 0.384·61-s − 1.95·67-s − 1.78·71-s + 1.63·73-s − 0.341·77-s + 1.68·79-s + 0.878·83-s + 0.953·89-s + 0.104·91-s + 0.507·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 93600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 3 T + p T^{2} \) |
| 67 | \( 1 + 16 T + p T^{2} \) |
| 71 | \( 1 + 15 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 15 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 - 5 T + p T^{2} \) |
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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
−13.97211165458689, −13.44083118440219, −13.23503525996073, −12.73133644779115, −11.92622849513588, −11.71004555540855, −11.05031882115713, −10.64793614592696, −10.24431945837897, −9.548027210324708, −9.109803253196272, −8.431394144203910, −8.201743988519547, −7.483590140771931, −7.070574226543153, −6.357496046794975, −6.027005266719518, −5.065454791560809, −4.913828552200843, −4.348942654558865, −3.391427725637547, −3.093743915681751, −2.218215493402361, −1.753942322422044, −0.8428177840159135, 0,
0.8428177840159135, 1.753942322422044, 2.218215493402361, 3.093743915681751, 3.391427725637547, 4.348942654558865, 4.913828552200843, 5.065454791560809, 6.027005266719518, 6.357496046794975, 7.070574226543153, 7.483590140771931, 8.201743988519547, 8.431394144203910, 9.109803253196272, 9.548027210324708, 10.24431945837897, 10.64793614592696, 11.05031882115713, 11.71004555540855, 11.92622849513588, 12.73133644779115, 13.23503525996073, 13.44083118440219, 13.97211165458689
