L(s) = 1 | − i·2-s + i·3-s − 4-s + (1 − 2i)5-s + 6-s + i·8-s − 9-s + (−2 − i)10-s − i·12-s − 2i·13-s + (2 + i)15-s + 16-s + 2i·17-s + i·18-s + 2·19-s + (−1 + 2i)20-s + ⋯ |
L(s) = 1 | − 0.707i·2-s + 0.577i·3-s − 0.5·4-s + (0.447 − 0.894i)5-s + 0.408·6-s + 0.353i·8-s − 0.333·9-s + (−0.632 − 0.316i)10-s − 0.288i·12-s − 0.554i·13-s + (0.516 + 0.258i)15-s + 0.250·16-s + 0.485i·17-s + 0.235i·18-s + 0.458·19-s + (−0.223 + 0.447i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.444211781\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.444211781\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 + (-1 + 2i)T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 - 2iT - 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 + 8iT - 23T^{2} \) |
| 29 | \( 1 - 8T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 + 6iT - 37T^{2} \) |
| 41 | \( 1 + 10T + 41T^{2} \) |
| 43 | \( 1 + 2iT - 43T^{2} \) |
| 47 | \( 1 - 6iT - 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 - 12T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + 14iT - 67T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 + 10iT - 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 - 12iT - 83T^{2} \) |
| 89 | \( 1 + 14T + 89T^{2} \) |
| 97 | \( 1 + 14iT - 97T^{2} \) |
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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
−9.353434673950186741501825360433, −8.577952003610346816454468393426, −8.105252529058410679088420808538, −6.66147214651667964574699566764, −5.64468713834121442748398412585, −4.92719653795454403056502388287, −4.17220307708890667225788901902, −3.10495908572285584295515131038, −1.98376918962552250048975071727, −0.60111634040376251457823421806,
1.44465338010747811144219153142, 2.74580131967526383533619143421, 3.72258312326547493989756657378, 5.08203269684506091718479567691, 5.78841169345574841325169294100, 6.78407442677367540423210949802, 7.07290660704053498239680768188, 7.981977415397311781779893080535, 8.861454167459039808761036671625, 9.729200764796119309181695624812