L(s) = 1 | + (−0.258 + 0.965i)3-s + (0.866 − 0.5i)5-s + (1.67 + 0.965i)7-s + (−0.866 − 0.499i)9-s + (0.258 + 0.965i)15-s + (−1.36 + 1.36i)21-s + (−0.258 − 0.448i)23-s + (0.499 − 0.866i)25-s + (0.707 − 0.707i)27-s + (−1.5 − 0.866i)29-s + 1.93·35-s + (0.866 − 0.5i)41-s + (−1.22 − 0.707i)43-s − 45-s + (−0.965 + 1.67i)47-s + ⋯ |
L(s) = 1 | + (−0.258 + 0.965i)3-s + (0.866 − 0.5i)5-s + (1.67 + 0.965i)7-s + (−0.866 − 0.499i)9-s + (0.258 + 0.965i)15-s + (−1.36 + 1.36i)21-s + (−0.258 − 0.448i)23-s + (0.499 − 0.866i)25-s + (0.707 − 0.707i)27-s + (−1.5 − 0.866i)29-s + 1.93·35-s + (0.866 − 0.5i)41-s + (−1.22 − 0.707i)43-s − 45-s + (−0.965 + 1.67i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.324534222\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.324534222\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.258 - 0.965i)T \) |
| 5 | \( 1 + (-0.866 + 0.5i)T \) |
good | 7 | \( 1 + (-1.67 - 0.965i)T + (0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 + (0.258 + 0.448i)T + (-0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 + (0.965 - 1.67i)T + (-0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.448 - 0.258i)T + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (0.258 - 0.448i)T + (-0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 + 1.73iT - T^{2} \) |
| 97 | \( 1 + (0.5 + 0.866i)T^{2} \) |
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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
−9.751703303180239346710706974917, −9.024240145352123593012057624207, −8.520783619045291450820696527877, −7.66013612367254048323206603963, −6.11642580712567300303923982635, −5.59356269824348408626749112142, −4.88154310023735498560337614810, −4.20161972083372861273093438185, −2.65421908336412069359542945427, −1.66540134627859796854188377713,
1.42480268940059944557533779177, 2.02791327080258271498199240111, 3.45663528421374328759718544996, 4.84343208215919759162490210942, 5.47112597487993143571775924379, 6.46874068511562255923846163148, 7.25578609310451640136417267234, 7.80946129109454345762452577932, 8.605912715807615447642674140764, 9.688427002007089333885626214029