| L(s) = 1 | − 2-s + 4-s − 2·5-s + 7-s − 8-s + 2·10-s − 6·13-s − 14-s + 16-s − 2·20-s − 8·23-s − 25-s + 6·26-s + 28-s − 6·29-s + 8·31-s − 32-s − 2·35-s − 10·37-s + 2·40-s − 6·41-s + 12·43-s + 8·46-s + 49-s + 50-s − 6·52-s + 10·53-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.894·5-s + 0.377·7-s − 0.353·8-s + 0.632·10-s − 1.66·13-s − 0.267·14-s + 1/4·16-s − 0.447·20-s − 1.66·23-s − 1/5·25-s + 1.17·26-s + 0.188·28-s − 1.11·29-s + 1.43·31-s − 0.176·32-s − 0.338·35-s − 1.64·37-s + 0.316·40-s − 0.937·41-s + 1.82·43-s + 1.17·46-s + 1/7·49-s + 0.141·50-s − 0.832·52-s + 1.37·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36414 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36414 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 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
−15.31072180149466, −14.77538633534627, −14.09664288025370, −13.85791964720222, −12.84536275081855, −12.32207116803839, −11.86185032292360, −11.65193548485167, −10.91784899352139, −10.23017735141638, −9.941772677643344, −9.351203735951188, −8.562693617366868, −8.209650172283849, −7.561911468472730, −7.299409952330433, −6.660550025629577, −5.808677275990514, −5.270529418982133, −4.493345610928792, −3.961736838358943, −3.259977078685670, −2.277277399242057, −1.980332611907489, −0.7289241704666713, 0,
0.7289241704666713, 1.980332611907489, 2.277277399242057, 3.259977078685670, 3.961736838358943, 4.493345610928792, 5.270529418982133, 5.808677275990514, 6.660550025629577, 7.299409952330433, 7.561911468472730, 8.209650172283849, 8.562693617366868, 9.351203735951188, 9.941772677643344, 10.23017735141638, 10.91784899352139, 11.65193548485167, 11.86185032292360, 12.32207116803839, 12.84536275081855, 13.85791964720222, 14.09664288025370, 14.77538633534627, 15.31072180149466
