L(s) = 1 | + i·3-s − i·7-s − 9-s − 6·11-s + 5i·13-s − 6i·17-s + 5·19-s + 21-s + 6i·23-s − i·27-s − 6·29-s − 31-s − 6i·33-s − 2i·37-s − 5·39-s + ⋯ |
L(s) = 1 | + 0.577i·3-s − 0.377i·7-s − 0.333·9-s − 1.80·11-s + 1.38i·13-s − 1.45i·17-s + 1.14·19-s + 0.218·21-s + 1.25i·23-s − 0.192i·27-s − 1.11·29-s − 0.179·31-s − 1.04i·33-s − 0.328i·37-s − 0.800·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9385921608\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9385921608\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
good | 7 | \( 1 + iT - 7T^{2} \) |
| 11 | \( 1 + 6T + 11T^{2} \) |
| 13 | \( 1 - 5iT - 13T^{2} \) |
| 17 | \( 1 + 6iT - 17T^{2} \) |
| 19 | \( 1 - 5T + 19T^{2} \) |
| 23 | \( 1 - 6iT - 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + T + 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + iT - 43T^{2} \) |
| 47 | \( 1 - 6iT - 47T^{2} \) |
| 53 | \( 1 + 12iT - 53T^{2} \) |
| 59 | \( 1 + 6T + 59T^{2} \) |
| 61 | \( 1 - 13T + 61T^{2} \) |
| 67 | \( 1 + 11iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 2iT - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 + 6iT - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 7iT - 97T^{2} \) |
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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
−8.020302983398762490374308801290, −7.42647442619073308913700446337, −6.94543886427299249698296477162, −5.65814781137762651310429143482, −5.23800960721941197514360612449, −4.52224322818064278823693697505, −3.57592167631857452520778280353, −2.83090353838038663148841739265, −1.83893467809733139436190253683, −0.29430172760439236458916789589,
0.935542377311310171069205104773, 2.25317848805833607774751430387, 2.83558226706090182571971106320, 3.74455416746621122666202831079, 4.98147381913464445755744155115, 5.59937913490160382905881450944, 6.01942560838314896012110546510, 7.16991709322426984358270463569, 7.74423522548539021008573965336, 8.263036196814095423426182857056