| L(s) = 1 | − 2-s + 4-s − 8-s − 3·11-s − 13-s + 16-s + 6·17-s + 19-s + 3·22-s + 9·23-s + 26-s − 6·29-s − 8·31-s − 32-s − 6·34-s + 7·37-s − 38-s + 3·41-s − 2·43-s − 3·44-s − 9·46-s − 9·47-s − 52-s + 9·53-s + 6·58-s − 8·61-s + 8·62-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s − 0.904·11-s − 0.277·13-s + 1/4·16-s + 1.45·17-s + 0.229·19-s + 0.639·22-s + 1.87·23-s + 0.196·26-s − 1.11·29-s − 1.43·31-s − 0.176·32-s − 1.02·34-s + 1.15·37-s − 0.162·38-s + 0.468·41-s − 0.304·43-s − 0.452·44-s − 1.32·46-s − 1.31·47-s − 0.138·52-s + 1.23·53-s + 0.787·58-s − 1.02·61-s + 1.01·62-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 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−16.09807199434801, −15.11501288176953, −14.82303627318169, −14.49810146422844, −13.40834029940265, −13.12311000433349, −12.53972283490055, −11.93056578159023, −11.25195537941704, −10.85471255846578, −10.28316804819862, −9.630786113134209, −9.257853165120536, −8.596262213086212, −7.848826901902556, −7.463283208243049, −7.046060190166271, −6.103661299854130, −5.443610148404079, −5.099282505911397, −4.084848068891306, −3.169881369202393, −2.810052832218957, −1.785466535068827, −1.026465001122381, 0,
1.026465001122381, 1.785466535068827, 2.810052832218957, 3.169881369202393, 4.084848068891306, 5.099282505911397, 5.443610148404079, 6.103661299854130, 7.046060190166271, 7.463283208243049, 7.848826901902556, 8.596262213086212, 9.257853165120536, 9.630786113134209, 10.28316804819862, 10.85471255846578, 11.25195537941704, 11.93056578159023, 12.53972283490055, 13.12311000433349, 13.40834029940265, 14.49810146422844, 14.82303627318169, 15.11501288176953, 16.09807199434801
