| L(s) = 1 | + 2-s + 4-s + 8-s − 2·11-s − 6·13-s + 16-s + 2·17-s − 2·22-s + 4·23-s − 6·26-s + 8·31-s + 32-s + 2·34-s + 2·37-s + 2·41-s − 4·43-s − 2·44-s + 4·46-s − 8·47-s − 6·52-s − 6·53-s + 10·59-s − 2·61-s + 8·62-s + 64-s − 8·67-s + 2·68-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.353·8-s − 0.603·11-s − 1.66·13-s + 1/4·16-s + 0.485·17-s − 0.426·22-s + 0.834·23-s − 1.17·26-s + 1.43·31-s + 0.176·32-s + 0.342·34-s + 0.328·37-s + 0.312·41-s − 0.609·43-s − 0.301·44-s + 0.589·46-s − 1.16·47-s − 0.832·52-s − 0.824·53-s + 1.30·59-s − 0.256·61-s + 1.01·62-s + 1/8·64-s − 0.977·67-s + 0.242·68-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 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 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 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.73325490934321, −15.09558050350882, −14.71035818241066, −14.29573078834552, −13.59306201601649, −13.03640401459112, −12.67824910124685, −11.93013673657478, −11.70935116312820, −10.92403241304703, −10.26827260156112, −9.875062416219078, −9.294585912850502, −8.368140847215170, −7.886394049132306, −7.245756480113507, −6.789322369642588, −6.013926517238104, −5.356029576944658, −4.775890226275020, −4.420779064646366, −3.315063518187133, −2.830381416305905, −2.202705037592629, −1.180200956639165, 0,
1.180200956639165, 2.202705037592629, 2.830381416305905, 3.315063518187133, 4.420779064646366, 4.775890226275020, 5.356029576944658, 6.013926517238104, 6.789322369642588, 7.245756480113507, 7.886394049132306, 8.368140847215170, 9.294585912850502, 9.875062416219078, 10.26827260156112, 10.92403241304703, 11.70935116312820, 11.93013673657478, 12.67824910124685, 13.03640401459112, 13.59306201601649, 14.29573078834552, 14.71035818241066, 15.09558050350882, 15.73325490934321
