| L(s) = 1 | − 2·5-s − 3·9-s + 2·13-s − 6·17-s + 8·19-s − 25-s − 6·29-s − 8·31-s − 2·37-s + 2·41-s + 4·43-s + 6·45-s + 8·47-s + 6·53-s − 6·61-s − 4·65-s − 4·67-s − 8·71-s + 10·73-s − 16·79-s + 9·81-s + 8·83-s + 12·85-s + 6·89-s − 16·95-s + 6·97-s + 101-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 9-s + 0.554·13-s − 1.45·17-s + 1.83·19-s − 1/5·25-s − 1.11·29-s − 1.43·31-s − 0.328·37-s + 0.312·41-s + 0.609·43-s + 0.894·45-s + 1.16·47-s + 0.824·53-s − 0.768·61-s − 0.496·65-s − 0.488·67-s − 0.949·71-s + 1.17·73-s − 1.80·79-s + 81-s + 0.878·83-s + 1.30·85-s + 0.635·89-s − 1.64·95-s + 0.609·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−14.97822137195201, −14.23188892477808, −13.86819752241102, −13.35656465880697, −12.80284113227273, −12.14141484508642, −11.62663857721105, −11.27462377850560, −10.94933970364976, −10.26465472616726, −9.442705507145547, −8.890252633008921, −8.775505252044841, −7.856929288159684, −7.408882342237967, −7.111867070753838, −6.049099837683110, −5.803221265557578, −5.094594572345357, −4.418753248483483, −3.661525480578208, −3.404035307126002, −2.527365855570781, −1.832762336945072, −0.7744045392985993, 0,
0.7744045392985993, 1.832762336945072, 2.527365855570781, 3.404035307126002, 3.661525480578208, 4.418753248483483, 5.094594572345357, 5.803221265557578, 6.049099837683110, 7.111867070753838, 7.408882342237967, 7.856929288159684, 8.775505252044841, 8.890252633008921, 9.442705507145547, 10.26465472616726, 10.94933970364976, 11.27462377850560, 11.62663857721105, 12.14141484508642, 12.80284113227273, 13.35656465880697, 13.86819752241102, 14.23188892477808, 14.97822137195201
