L(s) = 1 | + i·3-s + 4-s + 2i·7-s − 9-s + i·12-s + 16-s − 2·21-s − i·27-s + 2i·28-s − 36-s − i·37-s + i·48-s − 3·49-s − 2i·63-s + 64-s + ⋯ |
L(s) = 1 | + i·3-s + 4-s + 2i·7-s − 9-s + i·12-s + 16-s − 2·21-s − i·27-s + 2i·28-s − 36-s − i·37-s + i·48-s − 3·49-s − 2i·63-s + 64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.470457906\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.470457906\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
| 37 | \( 1 + iT \) |
good | 2 | \( 1 - T^{2} \) |
| 7 | \( 1 - 2iT - T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - 2iT - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + 2iT - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + T^{2} \) |
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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
−9.236109315718599368052337129613, −8.602331946771432747992462363482, −7.926774885568900980806729926140, −6.81553199700463675489344403482, −5.84238916896418778329887318492, −5.64374630926559220231968422013, −4.65772101760165123298306839935, −3.43357047848899628505736781922, −2.70226732970003992426595199101, −1.98548331413407362276180711636,
0.927613096332938647775603914877, 1.79715951110790924092822455953, 2.97620720409812134232405578495, 3.76385792528669482693733357439, 4.87907079122531678846271638823, 6.01407364121643980287606356339, 6.71445813245961913022405014925, 7.15985191432586972939484790405, 7.78169779447524737277763241323, 8.359436938150970297570699482944