L(s) = 1 | − 1.73i·5-s + (−2 + 1.73i)7-s − 3.46i·11-s + 3.46i·13-s + 5.19i·17-s + 2·19-s − 6.92i·23-s + 2.00·25-s − 6·29-s + 10·31-s + (2.99 + 3.46i)35-s − 11·37-s + 1.73i·41-s + 5.19i·43-s + 9·47-s + ⋯ |
L(s) = 1 | − 0.774i·5-s + (−0.755 + 0.654i)7-s − 1.04i·11-s + 0.960i·13-s + 1.26i·17-s + 0.458·19-s − 1.44i·23-s + 0.400·25-s − 1.11·29-s + 1.79·31-s + (0.507 + 0.585i)35-s − 1.80·37-s + 0.270i·41-s + 0.792i·43-s + 1.31·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.944 + 0.327i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.944 + 0.327i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.569335425\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.569335425\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (2 - 1.73i)T \) |
good | 5 | \( 1 + 1.73iT - 5T^{2} \) |
| 11 | \( 1 + 3.46iT - 11T^{2} \) |
| 13 | \( 1 - 3.46iT - 13T^{2} \) |
| 17 | \( 1 - 5.19iT - 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 + 6.92iT - 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 - 10T + 31T^{2} \) |
| 37 | \( 1 + 11T + 37T^{2} \) |
| 41 | \( 1 - 1.73iT - 41T^{2} \) |
| 43 | \( 1 - 5.19iT - 43T^{2} \) |
| 47 | \( 1 - 9T + 47T^{2} \) |
| 53 | \( 1 - 12T + 53T^{2} \) |
| 59 | \( 1 - 3T + 59T^{2} \) |
| 61 | \( 1 - 3.46iT - 61T^{2} \) |
| 67 | \( 1 - 3.46iT - 67T^{2} \) |
| 71 | \( 1 + 13.8iT - 71T^{2} \) |
| 73 | \( 1 - 6.92iT - 73T^{2} \) |
| 79 | \( 1 + 12.1iT - 79T^{2} \) |
| 83 | \( 1 - 15T + 83T^{2} \) |
| 89 | \( 1 - 6.92iT - 89T^{2} \) |
| 97 | \( 1 + 17.3iT - 97T^{2} \) |
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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
−8.710497640101815835512361123389, −8.275315006751149221921348984980, −7.05694132376657547494061318262, −6.28140314945068241546060607494, −5.76212709247606018635998308512, −4.81274265967626635064188154403, −3.95732015033460497752393201286, −3.06174565463975225853253250619, −2.00099562698686908988470418507, −0.72386858374236196768303796602,
0.793600136377040141638880833366, 2.30506852419882470703272665207, 3.18705157859581624604303721240, 3.83005413877166323974061444072, 4.99663902628604351174422545490, 5.66405210675233930591190299267, 6.80649704633291353046573366105, 7.19066119438422454931155318321, 7.69458897784687602651183754310, 8.889803006369873202301972598927