L(s) = 1 | + (−0.142 + 0.989i)2-s + (0.959 + 0.281i)3-s + (−0.959 − 0.281i)4-s + (−0.841 + 0.540i)5-s + (−0.415 + 0.909i)6-s + (0.415 − 0.909i)8-s + (0.841 + 0.540i)9-s + (−0.415 − 0.909i)10-s + (−0.841 − 0.540i)12-s + (−0.959 + 0.281i)15-s + (0.841 + 0.540i)16-s + (1.14 + 0.989i)17-s + (−0.654 + 0.755i)18-s + (−1.49 + 1.29i)19-s + (0.959 − 0.281i)20-s + ⋯ |
L(s) = 1 | + (−0.142 + 0.989i)2-s + (0.959 + 0.281i)3-s + (−0.959 − 0.281i)4-s + (−0.841 + 0.540i)5-s + (−0.415 + 0.909i)6-s + (0.415 − 0.909i)8-s + (0.841 + 0.540i)9-s + (−0.415 − 0.909i)10-s + (−0.841 − 0.540i)12-s + (−0.959 + 0.281i)15-s + (0.841 + 0.540i)16-s + (1.14 + 0.989i)17-s + (−0.654 + 0.755i)18-s + (−1.49 + 1.29i)19-s + (0.959 − 0.281i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.904 - 0.427i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.904 - 0.427i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.145479673\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.145479673\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.142 - 0.989i)T \) |
| 3 | \( 1 + (-0.959 - 0.281i)T \) |
| 5 | \( 1 + (0.841 - 0.540i)T \) |
| 23 | \( 1 + (-0.415 - 0.909i)T \) |
good | 7 | \( 1 + (0.959 + 0.281i)T^{2} \) |
| 11 | \( 1 + (-0.654 + 0.755i)T^{2} \) |
| 13 | \( 1 + (-0.959 + 0.281i)T^{2} \) |
| 17 | \( 1 + (-1.14 - 0.989i)T + (0.142 + 0.989i)T^{2} \) |
| 19 | \( 1 + (1.49 - 1.29i)T + (0.142 - 0.989i)T^{2} \) |
| 29 | \( 1 + (-0.142 - 0.989i)T^{2} \) |
| 31 | \( 1 + (0.557 + 1.89i)T + (-0.841 + 0.540i)T^{2} \) |
| 37 | \( 1 + (-0.415 - 0.909i)T^{2} \) |
| 41 | \( 1 + (-0.415 + 0.909i)T^{2} \) |
| 43 | \( 1 + (0.841 + 0.540i)T^{2} \) |
| 47 | \( 1 - 1.81iT - T^{2} \) |
| 53 | \( 1 + (0.118 - 0.822i)T + (-0.959 - 0.281i)T^{2} \) |
| 59 | \( 1 + (0.959 - 0.281i)T^{2} \) |
| 61 | \( 1 + (0.797 - 0.234i)T + (0.841 - 0.540i)T^{2} \) |
| 67 | \( 1 + (-0.654 - 0.755i)T^{2} \) |
| 71 | \( 1 + (-0.654 - 0.755i)T^{2} \) |
| 73 | \( 1 + (0.142 - 0.989i)T^{2} \) |
| 79 | \( 1 + (-0.239 - 1.66i)T + (-0.959 + 0.281i)T^{2} \) |
| 83 | \( 1 + (-0.304 + 0.474i)T + (-0.415 - 0.909i)T^{2} \) |
| 89 | \( 1 + (0.841 + 0.540i)T^{2} \) |
| 97 | \( 1 + (0.415 - 0.909i)T^{2} \) |
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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
−9.161557255522577448877599599227, −8.182740811508226688606705000207, −7.954987009679476639534230851501, −7.35277200744925337233082998802, −6.34856396972779026347242646011, −5.66643178565834826652918266507, −4.36423098320456532906987504319, −3.93980103252769648238073098827, −3.13210837930803430169337263902, −1.65451083409514520856966472759,
0.72336248364201634460489554317, 1.94783086268519644958478907886, 2.99928952622289030081039182984, 3.56825176072447047381846093171, 4.60905917487722493838506578869, 5.05014261860082559192158889870, 6.69725988745347679900401472926, 7.41229884713409024516965792764, 8.215730171886763424607457708577, 8.795308484794035009706655763161