L(s) = 1 | + (0.5 + 0.866i)4-s + (0.866 + 0.5i)5-s + (−1.22 + 0.707i)11-s + (−0.499 + 0.866i)16-s + (0.707 − 1.22i)19-s + 0.999i·20-s + (0.499 + 0.866i)25-s + 1.41i·29-s + (0.707 + 1.22i)31-s − 2i·41-s + (−1.22 − 0.707i)44-s − 1.41·55-s + (−1.73 + i)59-s + (0.707 − 1.22i)61-s − 0.999·64-s + ⋯ |
L(s) = 1 | + (0.5 + 0.866i)4-s + (0.866 + 0.5i)5-s + (−1.22 + 0.707i)11-s + (−0.499 + 0.866i)16-s + (0.707 − 1.22i)19-s + 0.999i·20-s + (0.499 + 0.866i)25-s + 1.41i·29-s + (0.707 + 1.22i)31-s − 2i·41-s + (−1.22 − 0.707i)44-s − 1.41·55-s + (−1.73 + i)59-s + (0.707 − 1.22i)61-s − 0.999·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.230 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.230 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.433112766\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.433112766\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (-0.866 - 0.5i)T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 - 1.41iT - T^{2} \) |
| 31 | \( 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 + 2iT - T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + (1.73 - i)T + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + 1.41iT - T^{2} \) |
| 73 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 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
−9.292277300726134724992852885597, −8.706868114371827435606473298255, −7.62311219716900660091413501379, −7.14688891774832914706057074019, −6.50149036955588833596493200405, −5.38270618981733550517822533105, −4.74344635636137458395767272607, −3.31914397779436588684965830905, −2.73465885250548505918796916047, −1.83892799412885731497627050875,
1.01045617439197845892669328208, 2.15016703787579882481094552182, 2.96700661919770309320746975930, 4.44583200133061544050205868686, 5.33836362212492671288798016620, 5.90652614232573802089730823295, 6.39618301287882738033445021272, 7.70629035312440233162786626268, 8.177366659674977095311488558965, 9.297009509168610561000865011847