L(s) = 1 | − 2-s + 4-s − 8-s − 3·11-s − 2·13-s + 16-s + 3·17-s + 7·19-s + 3·22-s + 2·26-s + 6·29-s + 4·31-s − 32-s − 3·34-s + 8·37-s − 7·38-s − 9·41-s + 8·43-s − 3·44-s − 6·47-s − 2·52-s + 12·53-s − 6·58-s + 12·59-s + 10·61-s − 4·62-s + 64-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s − 0.904·11-s − 0.554·13-s + 1/4·16-s + 0.727·17-s + 1.60·19-s + 0.639·22-s + 0.392·26-s + 1.11·29-s + 0.718·31-s − 0.176·32-s − 0.514·34-s + 1.31·37-s − 1.13·38-s − 1.40·41-s + 1.21·43-s − 0.452·44-s − 0.875·47-s − 0.277·52-s + 1.64·53-s − 0.787·58-s + 1.56·59-s + 1.28·61-s − 0.508·62-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
\(L(1)\) |
\(\approx\) |
\(1.573250152\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.573250152\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 5 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 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
−15.64189293208599, −15.08363902048432, −14.54553121385849, −13.91069442701851, −13.39266740668764, −12.77190711971305, −12.07185861557012, −11.72200549419279, −11.17865485767944, −10.28392825626033, −10.02940817296462, −9.640834611301561, −8.772278834099148, −8.259848080579670, −7.683542218873003, −7.242100271096504, −6.605275807537992, −5.727484980726217, −5.303684342074800, −4.610315145984566, −3.658621859228769, −2.832815335123097, −2.473698234812082, −1.295716646638888, −0.6238007583305811,
0.6238007583305811, 1.295716646638888, 2.473698234812082, 2.832815335123097, 3.658621859228769, 4.610315145984566, 5.303684342074800, 5.727484980726217, 6.605275807537992, 7.242100271096504, 7.683542218873003, 8.259848080579670, 8.772278834099148, 9.640834611301561, 10.02940817296462, 10.28392825626033, 11.17865485767944, 11.72200549419279, 12.07185861557012, 12.77190711971305, 13.39266740668764, 13.91069442701851, 14.54553121385849, 15.08363902048432, 15.64189293208599