L(s) = 1 | + i·3-s + i·5-s − 2·7-s − 9-s + 4i·11-s − 15-s − 6·17-s + 4i·19-s − 2i·21-s + 4·23-s − 25-s − i·27-s + 6i·29-s − 10·31-s − 4·33-s + ⋯ |
L(s) = 1 | + 0.577i·3-s + 0.447i·5-s − 0.755·7-s − 0.333·9-s + 1.20i·11-s − 0.258·15-s − 1.45·17-s + 0.917i·19-s − 0.436i·21-s + 0.834·23-s − 0.200·25-s − 0.192i·27-s + 1.11i·29-s − 1.79·31-s − 0.696·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 480 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.350034 + 0.845057i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.350034 + 0.845057i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 - iT \) |
good | 7 | \( 1 + 2T + 7T^{2} \) |
| 11 | \( 1 - 4iT - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 + 6T + 17T^{2} \) |
| 19 | \( 1 - 4iT - 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 - 6iT - 29T^{2} \) |
| 31 | \( 1 + 10T + 31T^{2} \) |
| 37 | \( 1 + 4iT - 37T^{2} \) |
| 41 | \( 1 - 10T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 - 4T + 47T^{2} \) |
| 53 | \( 1 - 10iT - 53T^{2} \) |
| 59 | \( 1 - 8iT - 59T^{2} \) |
| 61 | \( 1 - 8iT - 61T^{2} \) |
| 67 | \( 1 + 12iT - 67T^{2} \) |
| 71 | \( 1 - 4T + 71T^{2} \) |
| 73 | \( 1 - 10T + 73T^{2} \) |
| 79 | \( 1 - 14T + 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 - 14T + 89T^{2} \) |
| 97 | \( 1 + 10T + 97T^{2} \) |
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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
−10.96918157989956272974929345013, −10.57768525034885168649416590341, −9.433301718320780963657626634170, −9.023131166919874875061730635826, −7.51388341318594087755656447984, −6.79602086509534910021685230393, −5.70573512328631439719657914075, −4.50758744703912256829584128254, −3.52668710357401300722916867974, −2.21615428018457077783170334966,
0.53302017291401782773935868116, 2.39246308622132532311376050280, 3.63797042099591533503057469427, 5.01336486217339172908995425412, 6.14842491138407039726947953797, 6.83626671141431444491466164445, 7.993488914127447398452585955021, 8.923889297823320942192708446808, 9.490365955720418167979807092638, 11.01760013051485546990871307691