| L(s) = 1 | + (−0.146 − 1.40i)2-s + (1.20 − 1.00i)3-s + (−1.95 + 0.412i)4-s + (0.709 + 0.258i)5-s + (−1.59 − 1.54i)6-s + (−0.937 − 0.541i)7-s + (0.866 + 2.69i)8-s + (−0.0947 + 0.537i)9-s + (0.259 − 1.03i)10-s + (−2.64 + 1.52i)11-s + (−1.93 + 2.46i)12-s + (2.97 − 3.54i)13-s + (−0.624 + 1.39i)14-s + (1.11 − 0.404i)15-s + (3.66 − 1.61i)16-s + (0.837 + 4.75i)17-s + ⋯ |
| L(s) = 1 | + (−0.103 − 0.994i)2-s + (0.692 − 0.581i)3-s + (−0.978 + 0.206i)4-s + (0.317 + 0.115i)5-s + (−0.650 − 0.628i)6-s + (−0.354 − 0.204i)7-s + (0.306 + 0.951i)8-s + (−0.0315 + 0.179i)9-s + (0.0820 − 0.327i)10-s + (−0.798 + 0.460i)11-s + (−0.558 + 0.711i)12-s + (0.825 − 0.983i)13-s + (−0.166 + 0.373i)14-s + (0.287 − 0.104i)15-s + (0.915 − 0.403i)16-s + (0.203 + 1.15i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0902 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0902 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.731232 - 0.667948i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.731232 - 0.667948i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.146 + 1.40i)T \) |
| 19 | \( 1 + (-1.62 - 4.04i)T \) |
| good | 3 | \( 1 + (-1.20 + 1.00i)T + (0.520 - 2.95i)T^{2} \) |
| 5 | \( 1 + (-0.709 - 0.258i)T + (3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (0.937 + 0.541i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (2.64 - 1.52i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.97 + 3.54i)T + (-2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-0.837 - 4.75i)T + (-15.9 + 5.81i)T^{2} \) |
| 23 | \( 1 + (1.30 + 3.57i)T + (-17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (6.38 + 1.12i)T + (27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (0.249 - 0.431i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 2.70iT - 37T^{2} \) |
| 41 | \( 1 + (3.63 + 4.32i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-2.40 + 6.59i)T + (-32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (8.31 + 1.46i)T + (44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + (-4.64 - 12.7i)T + (-40.6 + 34.0i)T^{2} \) |
| 59 | \( 1 + (-0.185 - 1.05i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (-5.54 + 2.01i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (2.28 - 12.9i)T + (-62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (-2.22 - 0.810i)T + (54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + (-4.77 + 4.00i)T + (12.6 - 71.8i)T^{2} \) |
| 79 | \( 1 + (-10.7 + 8.98i)T + (13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-10.7 - 6.20i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-3.51 + 4.19i)T + (-15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (3.44 - 0.608i)T + (91.1 - 33.1i)T^{2} \) |
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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
−13.83776128719828355141772266594, −13.11934069652099949068923889505, −12.38519524314624761177672459300, −10.70618043597520280805054094755, −10.06598376664749676039648158561, −8.516650897769606244583690799066, −7.72590861513373747486267955359, −5.64958358190154016126938083436, −3.61301349341620332345831215365, −2.07652963311676610486041382551,
3.48065716452758005034365766710, 5.14294768992062653407364154922, 6.50812387634537586954372359575, 7.970263427864867504773412463442, 9.247872369671379469263250465543, 9.612381494870970173338897660633, 11.40214930081325635722250939155, 13.23653448004415687692540899683, 13.80887326188220737877592552927, 14.92187584057338931425450506533
