| L(s) = 1 | − 2-s + 4-s − 8-s + 4·11-s − 2·13-s + 16-s + 6·17-s − 4·22-s − 8·23-s + 2·26-s − 10·29-s + 8·31-s − 32-s − 6·34-s − 2·37-s − 2·41-s − 8·43-s + 4·44-s + 8·46-s − 4·47-s − 2·52-s + 10·53-s + 10·58-s + 4·59-s + 6·61-s − 8·62-s + 64-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s + 1.20·11-s − 0.554·13-s + 1/4·16-s + 1.45·17-s − 0.852·22-s − 1.66·23-s + 0.392·26-s − 1.85·29-s + 1.43·31-s − 0.176·32-s − 1.02·34-s − 0.328·37-s − 0.312·41-s − 1.21·43-s + 0.603·44-s + 1.17·46-s − 0.583·47-s − 0.277·52-s + 1.37·53-s + 1.31·58-s + 0.520·59-s + 0.768·61-s − 1.01·62-s + 1/8·64-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 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 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
−16.02878214727911, −15.20122094143928, −14.70467125109368, −14.39760714272380, −13.68482520094834, −13.10820201871534, −12.23283715402828, −11.83710478996501, −11.69033058600756, −10.73503789652551, −10.12950840751589, −9.670320269718557, −9.374151505189798, −8.342631946456222, −8.194703143229895, −7.367877437769802, −6.885119504131179, −6.185658192912122, −5.632061781525228, −4.927811126305924, −3.864502869470849, −3.609952139471902, −2.551925596272550, −1.788526654385696, −1.089197585588818, 0,
1.089197585588818, 1.788526654385696, 2.551925596272550, 3.609952139471902, 3.864502869470849, 4.927811126305924, 5.632061781525228, 6.185658192912122, 6.885119504131179, 7.367877437769802, 8.194703143229895, 8.342631946456222, 9.374151505189798, 9.670320269718557, 10.12950840751589, 10.73503789652551, 11.69033058600756, 11.83710478996501, 12.23283715402828, 13.10820201871534, 13.68482520094834, 14.39760714272380, 14.70467125109368, 15.20122094143928, 16.02878214727911
