| L(s) = 1 | − 3·9-s − 2·11-s − 7·17-s − 3·23-s − 6·29-s + 7·31-s + 4·37-s − 7·41-s − 8·43-s + 7·47-s − 4·53-s − 14·59-s + 14·61-s + 12·67-s + 71-s + 14·73-s + 11·79-s + 9·81-s + 14·83-s + 7·89-s − 7·97-s + 6·99-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 9-s − 0.603·11-s − 1.69·17-s − 0.625·23-s − 1.11·29-s + 1.25·31-s + 0.657·37-s − 1.09·41-s − 1.21·43-s + 1.02·47-s − 0.549·53-s − 1.82·59-s + 1.79·61-s + 1.46·67-s + 0.118·71-s + 1.63·73-s + 1.23·79-s + 81-s + 1.53·83-s + 0.741·89-s − 0.710·97-s + 0.603·99-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 7 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 - 7 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−14.15690664463300, −13.65891636587168, −13.41691773631184, −12.87029265460694, −12.16432917165764, −11.80702904264485, −11.19573100538473, −10.85487708396305, −10.40376050956771, −9.529204229439644, −9.375964242405938, −8.605684840824361, −8.098440930565484, −7.924822929371540, −6.930898487913833, −6.541587644594721, −6.078098287208241, −5.282160097832501, −5.008228221030165, −4.236177715148181, −3.651263557151073, −2.960217864110551, −2.290436649349630, −1.928084130098106, −0.7003823003846423, 0,
0.7003823003846423, 1.928084130098106, 2.290436649349630, 2.960217864110551, 3.651263557151073, 4.236177715148181, 5.008228221030165, 5.282160097832501, 6.078098287208241, 6.541587644594721, 6.930898487913833, 7.924822929371540, 8.098440930565484, 8.605684840824361, 9.375964242405938, 9.529204229439644, 10.40376050956771, 10.85487708396305, 11.19573100538473, 11.80702904264485, 12.16432917165764, 12.87029265460694, 13.41691773631184, 13.65891636587168, 14.15690664463300
