| L(s) = 1 | + 5-s − 4·7-s − 2·11-s − 2·17-s + 23-s + 25-s + 4·29-s − 4·35-s + 10·37-s − 6·41-s − 2·43-s + 12·47-s + 9·49-s − 6·53-s − 2·55-s + 12·59-s − 14·61-s − 2·67-s − 2·71-s + 6·73-s + 8·77-s − 8·79-s + 8·83-s − 2·85-s + 8·89-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.51·7-s − 0.603·11-s − 0.485·17-s + 0.208·23-s + 1/5·25-s + 0.742·29-s − 0.676·35-s + 1.64·37-s − 0.937·41-s − 0.304·43-s + 1.75·47-s + 9/7·49-s − 0.824·53-s − 0.269·55-s + 1.56·59-s − 1.79·61-s − 0.244·67-s − 0.237·71-s + 0.702·73-s + 0.911·77-s − 0.900·79-s + 0.878·83-s − 0.216·85-s + 0.847·89-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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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
−16.16114781584745, −15.66873351720695, −15.25149443473110, −14.53041410753484, −13.82986050390335, −13.31371700862716, −13.01533014922645, −12.41454792896688, −11.84150115839381, −11.06234562827042, −10.39023492842825, −10.08040196290664, −9.345243042901177, −9.016782383810490, −8.206110771445664, −7.501873335366010, −6.812560560697519, −6.298761683893472, −5.810568695681951, −5.012476908965678, −4.290874794564692, −3.448030102845538, −2.801102523203020, −2.237452921737085, −0.9995310421551456, 0,
0.9995310421551456, 2.237452921737085, 2.801102523203020, 3.448030102845538, 4.290874794564692, 5.012476908965678, 5.810568695681951, 6.298761683893472, 6.812560560697519, 7.501873335366010, 8.206110771445664, 9.016782383810490, 9.345243042901177, 10.08040196290664, 10.39023492842825, 11.06234562827042, 11.84150115839381, 12.41454792896688, 13.01533014922645, 13.31371700862716, 13.82986050390335, 14.53041410753484, 15.25149443473110, 15.66873351720695, 16.16114781584745
