| L(s) = 1 | + 2-s + 4-s + 8-s − 2·11-s + 2·13-s + 16-s − 2·17-s − 6·19-s − 2·22-s − 4·23-s + 2·26-s + 2·31-s + 32-s − 2·34-s − 2·37-s − 6·38-s + 10·41-s + 8·43-s − 2·44-s − 4·46-s + 8·47-s + 2·52-s − 2·53-s + 4·59-s − 8·61-s + 2·62-s + 64-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.353·8-s − 0.603·11-s + 0.554·13-s + 1/4·16-s − 0.485·17-s − 1.37·19-s − 0.426·22-s − 0.834·23-s + 0.392·26-s + 0.359·31-s + 0.176·32-s − 0.342·34-s − 0.328·37-s − 0.973·38-s + 1.56·41-s + 1.21·43-s − 0.301·44-s − 0.589·46-s + 1.16·47-s + 0.277·52-s − 0.274·53-s + 0.520·59-s − 1.02·61-s + 0.254·62-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 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−15.63249228869362, −15.44494014511506, −14.56778617781014, −14.20538597851438, −13.65870135044399, −12.99929821205196, −12.70288359460159, −12.14781707152569, −11.37449129445990, −10.97708871994197, −10.40573082947548, −9.954790194394287, −8.887896977874266, −8.717820684687492, −7.733470020749383, −7.460338167161260, −6.487447668125782, −6.100006235901599, −5.571676768576951, −4.677641478971687, −4.209731327973886, −3.638133563884158, −2.599002685301950, −2.261352901383471, −1.202427350110003, 0,
1.202427350110003, 2.261352901383471, 2.599002685301950, 3.638133563884158, 4.209731327973886, 4.677641478971687, 5.571676768576951, 6.100006235901599, 6.487447668125782, 7.460338167161260, 7.733470020749383, 8.717820684687492, 8.887896977874266, 9.954790194394287, 10.40573082947548, 10.97708871994197, 11.37449129445990, 12.14781707152569, 12.70288359460159, 12.99929821205196, 13.65870135044399, 14.20538597851438, 14.56778617781014, 15.44494014511506, 15.63249228869362
