L(s) = 1 | − 2-s + 4-s − 8-s − 6·11-s − 13-s + 16-s − 3·17-s + 4·19-s + 6·22-s − 3·23-s + 26-s − 3·29-s − 5·31-s − 32-s + 3·34-s + 10·37-s − 4·38-s + 9·41-s + 43-s − 6·44-s + 3·46-s − 52-s + 9·53-s + 3·58-s + 9·59-s − 11·61-s + 5·62-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s − 1.80·11-s − 0.277·13-s + 1/4·16-s − 0.727·17-s + 0.917·19-s + 1.27·22-s − 0.625·23-s + 0.196·26-s − 0.557·29-s − 0.898·31-s − 0.176·32-s + 0.514·34-s + 1.64·37-s − 0.648·38-s + 1.40·41-s + 0.152·43-s − 0.904·44-s + 0.442·46-s − 0.138·52-s + 1.23·53-s + 0.393·58-s + 1.17·59-s − 1.40·61-s + 0.635·62-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 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 + 11 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−15.85479185296458, −15.47681204276855, −14.81767212795085, −14.27389655476537, −13.54450754443294, −12.97782807184170, −12.68594666091630, −11.86583963155344, −11.18083182028405, −10.96695133243986, −10.13182794161053, −9.844031111749581, −9.162046507577586, −8.554666194643972, −7.894117135360238, −7.456283844116017, −7.074136183766714, −5.930662369944489, −5.721768805811693, −4.889553218425689, −4.193151513610464, −3.252260901953890, −2.510726334924344, −2.082400114822403, −0.8721820604354980, 0,
0.8721820604354980, 2.082400114822403, 2.510726334924344, 3.252260901953890, 4.193151513610464, 4.889553218425689, 5.721768805811693, 5.930662369944489, 7.074136183766714, 7.456283844116017, 7.894117135360238, 8.554666194643972, 9.162046507577586, 9.844031111749581, 10.13182794161053, 10.96695133243986, 11.18083182028405, 11.86583963155344, 12.68594666091630, 12.97782807184170, 13.54450754443294, 14.27389655476537, 14.81767212795085, 15.47681204276855, 15.85479185296458