| L(s) = 1 | − 3·7-s + 11-s + 13-s − 3·17-s − 4·19-s + 2·23-s − 3·29-s + 5·31-s + 2·37-s + 2·43-s + 9·47-s + 2·49-s − 5·53-s + 7·59-s − 11·61-s + 3·67-s − 8·71-s + 2·73-s − 3·77-s + 10·79-s − 3·83-s + 8·89-s − 3·91-s − 6·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 1.13·7-s + 0.301·11-s + 0.277·13-s − 0.727·17-s − 0.917·19-s + 0.417·23-s − 0.557·29-s + 0.898·31-s + 0.328·37-s + 0.304·43-s + 1.31·47-s + 2/7·49-s − 0.686·53-s + 0.911·59-s − 1.40·61-s + 0.366·67-s − 0.949·71-s + 0.234·73-s − 0.341·77-s + 1.12·79-s − 0.329·83-s + 0.847·89-s − 0.314·91-s − 0.609·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 + 5 T + p T^{2} \) |
| 59 | \( 1 - 7 T + p T^{2} \) |
| 61 | \( 1 + 11 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−14.91153789215916, −14.36207847523483, −13.63521891577981, −13.38523449848415, −12.75679229702571, −12.44713265171731, −11.78375650718185, −11.19303949172382, −10.68152304891120, −10.22202002010914, −9.548011306392888, −9.113557920667118, −8.703451199610092, −7.987780126541914, −7.380086164038397, −6.651492147541725, −6.407594258487860, −5.845432727349704, −5.091379398588497, −4.305947074230763, −3.930883029287453, −3.132569157617583, −2.591848656027803, −1.830171004503181, −0.8546117989199394, 0,
0.8546117989199394, 1.830171004503181, 2.591848656027803, 3.132569157617583, 3.930883029287453, 4.305947074230763, 5.091379398588497, 5.845432727349704, 6.407594258487860, 6.651492147541725, 7.380086164038397, 7.987780126541914, 8.703451199610092, 9.113557920667118, 9.548011306392888, 10.22202002010914, 10.68152304891120, 11.19303949172382, 11.78375650718185, 12.44713265171731, 12.75679229702571, 13.38523449848415, 13.63521891577981, 14.36207847523483, 14.91153789215916
