| L(s) = 1 | − 2·3-s − 3·5-s − 3·7-s + 9-s − 6·11-s − 6·13-s + 6·15-s − 3·17-s − 8·19-s + 6·21-s − 2·23-s + 4·25-s + 4·27-s − 9·29-s + 2·31-s + 12·33-s + 9·35-s − 8·37-s + 12·39-s − 4·41-s + 43-s − 3·45-s − 3·47-s + 2·49-s + 6·51-s − 9·53-s + 18·55-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 1.34·5-s − 1.13·7-s + 1/3·9-s − 1.80·11-s − 1.66·13-s + 1.54·15-s − 0.727·17-s − 1.83·19-s + 1.30·21-s − 0.417·23-s + 4/5·25-s + 0.769·27-s − 1.67·29-s + 0.359·31-s + 2.08·33-s + 1.52·35-s − 1.31·37-s + 1.92·39-s − 0.624·41-s + 0.152·43-s − 0.447·45-s − 0.437·47-s + 2/7·49-s + 0.840·51-s − 1.23·53-s + 2.42·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30064 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30064 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 1879 | \( 1 + T \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 + 3 T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - T + p T^{2} \) |
| 71 | \( 1 + 5 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 7 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
| 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
−15.86966113094814, −15.59642688925967, −15.01777324235975, −14.62608308939919, −13.59392843432258, −12.96437709044100, −12.67061263990843, −12.25294656601051, −11.73838258064040, −11.00155245927683, −10.77942255826542, −10.13998467566540, −9.698082134149390, −8.772947488572741, −8.238554255806678, −7.608684996887939, −7.104925952872932, −6.597842723215102, −5.910195151979466, −5.282049768068779, −4.699876282598152, −4.232775554734428, −3.330310222054931, −2.707085497629219, −1.976628029882896, 0, 0, 0,
1.976628029882896, 2.707085497629219, 3.330310222054931, 4.232775554734428, 4.699876282598152, 5.282049768068779, 5.910195151979466, 6.597842723215102, 7.104925952872932, 7.608684996887939, 8.238554255806678, 8.772947488572741, 9.698082134149390, 10.13998467566540, 10.77942255826542, 11.00155245927683, 11.73838258064040, 12.25294656601051, 12.67061263990843, 12.96437709044100, 13.59392843432258, 14.62608308939919, 15.01777324235975, 15.59642688925967, 15.86966113094814
