| L(s) = 1 | − 2-s + 4-s − 8-s − 6·11-s − 6·13-s + 16-s − 4·19-s + 6·22-s + 6·26-s + 8·29-s + 2·31-s − 32-s − 4·37-s + 4·38-s + 10·41-s + 6·43-s − 6·44-s − 2·47-s − 6·52-s + 10·53-s − 8·58-s + 4·59-s + 14·61-s − 2·62-s + 64-s − 14·67-s − 8·71-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s − 1.80·11-s − 1.66·13-s + 1/4·16-s − 0.917·19-s + 1.27·22-s + 1.17·26-s + 1.48·29-s + 0.359·31-s − 0.176·32-s − 0.657·37-s + 0.648·38-s + 1.56·41-s + 0.914·43-s − 0.904·44-s − 0.291·47-s − 0.832·52-s + 1.37·53-s − 1.05·58-s + 0.520·59-s + 1.79·61-s − 0.254·62-s + 1/8·64-s − 1.71·67-s − 0.949·71-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 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 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
−15.84839557975836, −15.45464166986463, −14.65409533412423, −14.47169698835944, −13.55827829653476, −12.94613750313579, −12.56839546841186, −11.96357365126579, −11.39976029279770, −10.58160451613784, −10.18290179015451, −10.03843265647018, −9.039104770418424, −8.587484571715563, −7.944485401260118, −7.400586378497669, −7.052788384892586, −6.117428257102935, −5.555265842437234, −4.804568297362504, −4.362821647127738, −3.139344008754616, −2.466767582021363, −2.206600097071353, −0.8145461663546852, 0,
0.8145461663546852, 2.206600097071353, 2.466767582021363, 3.139344008754616, 4.362821647127738, 4.804568297362504, 5.555265842437234, 6.117428257102935, 7.052788384892586, 7.400586378497669, 7.944485401260118, 8.587484571715563, 9.039104770418424, 10.03843265647018, 10.18290179015451, 10.58160451613784, 11.39976029279770, 11.96357365126579, 12.56839546841186, 12.94613750313579, 13.55827829653476, 14.47169698835944, 14.65409533412423, 15.45464166986463, 15.84839557975836
