L(s) = 1 | − 2-s + 4-s − 8-s + 2·13-s + 16-s − 6·17-s + 4·19-s − 2·26-s + 6·29-s − 8·31-s − 32-s + 6·34-s − 2·37-s − 4·38-s − 6·41-s + 4·43-s + 2·52-s − 6·53-s − 6·58-s + 10·61-s + 8·62-s + 64-s + 4·67-s − 6·68-s + 2·73-s + 2·74-s + 4·76-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s + 0.554·13-s + 1/4·16-s − 1.45·17-s + 0.917·19-s − 0.392·26-s + 1.11·29-s − 1.43·31-s − 0.176·32-s + 1.02·34-s − 0.328·37-s − 0.648·38-s − 0.937·41-s + 0.609·43-s + 0.277·52-s − 0.824·53-s − 0.787·58-s + 1.28·61-s + 1.01·62-s + 1/8·64-s + 0.488·67-s − 0.727·68-s + 0.234·73-s + 0.232·74-s + 0.458·76-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 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 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−15.88581135912166, −15.56890959453130, −14.69234161069474, −14.31763120278503, −13.51745883228419, −13.19429444674155, −12.46652782485340, −11.85047537157887, −11.34318128686977, −10.80846524753785, −10.37126213569507, −9.605264836975086, −9.126737554498668, −8.638953323680762, −8.048251448656477, −7.415055999644690, −6.722798758963373, −6.395967763281527, −5.490145999878676, −4.948353104318621, −4.046513072371710, −3.410992455474333, −2.578757853491350, −1.859270346693527, −1.021439153079645, 0,
1.021439153079645, 1.859270346693527, 2.578757853491350, 3.410992455474333, 4.046513072371710, 4.948353104318621, 5.490145999878676, 6.395967763281527, 6.722798758963373, 7.415055999644690, 8.048251448656477, 8.638953323680762, 9.126737554498668, 9.605264836975086, 10.37126213569507, 10.80846524753785, 11.34318128686977, 11.85047537157887, 12.46652782485340, 13.19429444674155, 13.51745883228419, 14.31763120278503, 14.69234161069474, 15.56890959453130, 15.88581135912166