L(s) = 1 | + (−0.780 + 1.54i)3-s − 3.02·5-s − 2.62i·7-s + (−1.78 − 2.41i)9-s − 1.32·11-s − 1.56·13-s + (2.35 − 4.66i)15-s − 0.371·17-s − 6.71i·19-s + (4.05 + 2.04i)21-s + (−4.71 − 0.868i)23-s + 4.12·25-s + (5.12 − 0.868i)27-s + 4.82i·29-s − 8.68·31-s + ⋯ |
L(s) = 1 | + (−0.450 + 0.892i)3-s − 1.35·5-s − 0.991i·7-s + (−0.593 − 0.804i)9-s − 0.399·11-s − 0.433·13-s + (0.608 − 1.20i)15-s − 0.0901·17-s − 1.54i·19-s + (0.884 + 0.446i)21-s + (−0.983 − 0.181i)23-s + 0.824·25-s + (0.985 − 0.167i)27-s + 0.896i·29-s − 1.55·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.604 + 0.796i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.604 + 0.796i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0954294 - 0.192341i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0954294 - 0.192341i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.780 - 1.54i)T \) |
| 23 | \( 1 + (4.71 + 0.868i)T \) |
good | 5 | \( 1 + 3.02T + 5T^{2} \) |
| 7 | \( 1 + 2.62iT - 7T^{2} \) |
| 11 | \( 1 + 1.32T + 11T^{2} \) |
| 13 | \( 1 + 1.56T + 13T^{2} \) |
| 17 | \( 1 + 0.371T + 17T^{2} \) |
| 19 | \( 1 + 6.71iT - 19T^{2} \) |
| 29 | \( 1 - 4.82iT - 29T^{2} \) |
| 31 | \( 1 + 8.68T + 31T^{2} \) |
| 37 | \( 1 - 9.33iT - 37T^{2} \) |
| 41 | \( 1 + 1.35iT - 41T^{2} \) |
| 43 | \( 1 - 6.71iT - 43T^{2} \) |
| 47 | \( 1 + 11.0iT - 47T^{2} \) |
| 53 | \( 1 + 11.1T + 53T^{2} \) |
| 59 | \( 1 + 6.18iT - 59T^{2} \) |
| 61 | \( 1 - 5.24iT - 61T^{2} \) |
| 67 | \( 1 - 7.86iT - 67T^{2} \) |
| 71 | \( 1 - 3.09iT - 71T^{2} \) |
| 73 | \( 1 - 6.68T + 73T^{2} \) |
| 79 | \( 1 + 11.9iT - 79T^{2} \) |
| 83 | \( 1 - 10.7T + 83T^{2} \) |
| 89 | \( 1 - 7.73T + 89T^{2} \) |
| 97 | \( 1 + 4.09iT - 97T^{2} \) |
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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
−11.35239962002336340253758030778, −10.78019862668666719438561410571, −9.838285697066832278843724521254, −8.685900057169405235489777605990, −7.61152370054622109518482442083, −6.72084707805649956955986453113, −5.07445497586325710217459304795, −4.26048010384683140462395635308, −3.28246583349053593469215444275, −0.16515014063382521903013245421,
2.17501141909324510879865115537, 3.80070058080213375694752293376, 5.31009214935167703773922651489, 6.22767989751893403068053599893, 7.74729360814220164773547792804, 7.85520590417978622887019883657, 9.192699676600738319044661472841, 10.65553698030595159718957381016, 11.52977770673271560380870505767, 12.32523765659330054166375150488