| L(s) = 1 | + 2·7-s − 4·11-s − 13-s + 8·17-s − 6·19-s − 6·23-s − 4·29-s − 2·37-s + 2·41-s + 4·43-s − 3·49-s + 10·53-s − 4·59-s + 10·61-s − 12·67-s − 8·71-s + 8·73-s − 8·77-s − 8·79-s + 12·83-s + 14·89-s − 2·91-s + 16·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | + 0.755·7-s − 1.20·11-s − 0.277·13-s + 1.94·17-s − 1.37·19-s − 1.25·23-s − 0.742·29-s − 0.328·37-s + 0.312·41-s + 0.609·43-s − 3/7·49-s + 1.37·53-s − 0.520·59-s + 1.28·61-s − 1.46·67-s − 0.949·71-s + 0.936·73-s − 0.911·77-s − 0.900·79-s + 1.31·83-s + 1.48·89-s − 0.209·91-s + 1.62·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 187200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 187200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 8 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.36532249192694, −12.78367215171711, −12.49696952407427, −11.87722235141943, −11.64410390367267, −10.86729607670582, −10.47347592258729, −10.22019299367690, −9.677312771738294, −9.059074635035213, −8.444551287560496, −8.079783512126297, −7.563821699263142, −7.435819750005909, −6.521461009316223, −5.943983093258525, −5.532542878031436, −5.110151511716627, −4.502206901788876, −3.945680717199900, −3.399622502409603, −2.671780838908629, −2.139790484097724, −1.633114571899195, −0.7699884000748574, 0,
0.7699884000748574, 1.633114571899195, 2.139790484097724, 2.671780838908629, 3.399622502409603, 3.945680717199900, 4.502206901788876, 5.110151511716627, 5.532542878031436, 5.943983093258525, 6.521461009316223, 7.435819750005909, 7.563821699263142, 8.079783512126297, 8.444551287560496, 9.059074635035213, 9.677312771738294, 10.22019299367690, 10.47347592258729, 10.86729607670582, 11.64410390367267, 11.87722235141943, 12.49696952407427, 12.78367215171711, 13.36532249192694
