| L(s) = 1 | + 2·7-s − 11-s + 2·13-s + 2·19-s − 8·31-s + 2·37-s − 2·43-s − 3·49-s − 6·53-s + 12·59-s − 2·61-s + 4·67-s − 2·73-s − 2·77-s + 10·79-s − 12·83-s + 6·89-s + 4·91-s − 14·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 0.755·7-s − 0.301·11-s + 0.554·13-s + 0.458·19-s − 1.43·31-s + 0.328·37-s − 0.304·43-s − 3/7·49-s − 0.824·53-s + 1.56·59-s − 0.256·61-s + 0.488·67-s − 0.234·73-s − 0.227·77-s + 1.12·79-s − 1.31·83-s + 0.635·89-s + 0.419·91-s − 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 158400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 158400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−13.59512017487343, −12.95784490727470, −12.69614148590485, −12.07401186622164, −11.50205747931456, −11.14828416063797, −10.84630291632105, −10.16872300214008, −9.724079196971741, −9.190829631596118, −8.656852055485580, −8.212818957843826, −7.740158529499181, −7.283736298690915, −6.693065180237856, −6.166324685490848, −5.420232382712097, −5.275207615946607, −4.548292388285912, −3.989362929144147, −3.442727027693282, −2.838937889857522, −2.089278126934514, −1.578529762325916, −0.9063831822515350, 0,
0.9063831822515350, 1.578529762325916, 2.089278126934514, 2.838937889857522, 3.442727027693282, 3.989362929144147, 4.548292388285912, 5.275207615946607, 5.420232382712097, 6.166324685490848, 6.693065180237856, 7.283736298690915, 7.740158529499181, 8.212818957843826, 8.656852055485580, 9.190829631596118, 9.724079196971741, 10.16872300214008, 10.84630291632105, 11.14828416063797, 11.50205747931456, 12.07401186622164, 12.69614148590485, 12.95784490727470, 13.59512017487343
