| L(s) = 1 | − 2-s + 4-s − 8-s − 3·11-s − 4·13-s + 16-s + 4·19-s + 3·22-s + 4·26-s − 9·29-s + 31-s − 32-s − 8·37-s − 4·38-s + 10·43-s − 3·44-s + 6·47-s − 4·52-s − 3·53-s + 9·58-s + 3·59-s + 10·61-s − 62-s + 64-s + 10·67-s + 6·71-s + 2·73-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s − 0.904·11-s − 1.10·13-s + 1/4·16-s + 0.917·19-s + 0.639·22-s + 0.784·26-s − 1.67·29-s + 0.179·31-s − 0.176·32-s − 1.31·37-s − 0.648·38-s + 1.52·43-s − 0.452·44-s + 0.875·47-s − 0.554·52-s − 0.412·53-s + 1.18·58-s + 0.390·59-s + 1.28·61-s − 0.127·62-s + 1/8·64-s + 1.22·67-s + 0.712·71-s + 0.234·73-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 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−15.77365571179634, −15.47742340069390, −14.76973306206468, −14.27342621426990, −13.67467471236982, −12.98851745835964, −12.46051400172994, −12.00509756603300, −11.27953972429948, −10.84288544127910, −10.22050708158261, −9.669802472220123, −9.280856304148471, −8.568885684982358, −7.883770759217941, −7.407665731058286, −7.057296353492464, −6.170003955453934, −5.276780492663500, −5.214827680536388, −4.055225528412655, −3.352810968293428, −2.489852589587271, −2.044814388013950, −0.9051108715066395, 0,
0.9051108715066395, 2.044814388013950, 2.489852589587271, 3.352810968293428, 4.055225528412655, 5.214827680536388, 5.276780492663500, 6.170003955453934, 7.057296353492464, 7.407665731058286, 7.883770759217941, 8.568885684982358, 9.280856304148471, 9.669802472220123, 10.22050708158261, 10.84288544127910, 11.27953972429948, 12.00509756603300, 12.46051400172994, 12.98851745835964, 13.67467471236982, 14.27342621426990, 14.76973306206468, 15.47742340069390, 15.77365571179634
