L(s) = 1 | + (0.819 − 0.573i)2-s + (2.17 − 0.190i)3-s + (0.342 − 0.939i)4-s + (1.67 − 1.40i)6-s + (0.431 + 0.115i)7-s + (−0.258 − 0.965i)8-s + (1.74 − 0.307i)9-s + (2.37 − 4.12i)11-s + (0.565 − 2.11i)12-s + (0.210 − 2.40i)13-s + (0.419 − 0.152i)14-s + (−0.766 − 0.642i)16-s + (0.392 + 0.560i)17-s + (1.25 − 1.25i)18-s + (−2.15 + 3.79i)19-s + ⋯ |
L(s) = 1 | + (0.579 − 0.405i)2-s + (1.25 − 0.109i)3-s + (0.171 − 0.469i)4-s + (0.683 − 0.573i)6-s + (0.163 + 0.0436i)7-s + (−0.0915 − 0.341i)8-s + (0.582 − 0.102i)9-s + (0.717 − 1.24i)11-s + (0.163 − 0.609i)12-s + (0.0582 − 0.666i)13-s + (0.112 − 0.0408i)14-s + (−0.191 − 0.160i)16-s + (0.0951 + 0.135i)17-s + (0.295 − 0.295i)18-s + (−0.493 + 0.869i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.85786 - 1.76476i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.85786 - 1.76476i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.819 + 0.573i)T \) |
| 5 | \( 1 \) |
| 19 | \( 1 + (2.15 - 3.79i)T \) |
good | 3 | \( 1 + (-2.17 + 0.190i)T + (2.95 - 0.520i)T^{2} \) |
| 7 | \( 1 + (-0.431 - 0.115i)T + (6.06 + 3.5i)T^{2} \) |
| 11 | \( 1 + (-2.37 + 4.12i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.210 + 2.40i)T + (-12.8 - 2.25i)T^{2} \) |
| 17 | \( 1 + (-0.392 - 0.560i)T + (-5.81 + 15.9i)T^{2} \) |
| 23 | \( 1 + (0.188 + 0.404i)T + (-14.7 + 17.6i)T^{2} \) |
| 29 | \( 1 + (-0.892 - 5.06i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (-2.27 + 1.31i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.48 - 1.48i)T + 37iT^{2} \) |
| 41 | \( 1 + (0.333 - 0.397i)T + (-7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (-8.46 - 3.94i)T + (27.6 + 32.9i)T^{2} \) |
| 47 | \( 1 + (-4.32 - 3.02i)T + (16.0 + 44.1i)T^{2} \) |
| 53 | \( 1 + (-7.35 + 3.42i)T + (34.0 - 40.6i)T^{2} \) |
| 59 | \( 1 + (0.0555 - 0.315i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (7.77 + 2.82i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (6.44 - 9.20i)T + (-22.9 - 62.9i)T^{2} \) |
| 71 | \( 1 + (-0.879 - 2.41i)T + (-54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + (-0.378 - 4.32i)T + (-71.8 + 12.6i)T^{2} \) |
| 79 | \( 1 + (12.3 + 10.3i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-0.224 + 0.838i)T + (-71.8 - 41.5i)T^{2} \) |
| 89 | \( 1 + (12.7 - 10.7i)T + (15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (-2.46 + 1.72i)T + (33.1 - 91.1i)T^{2} \) |
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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
−9.891236630754399912792632769160, −8.904666366087740576611020452786, −8.391039092738831583407035768655, −7.54805600212219886963162796820, −6.30956348326414325520818717936, −5.56286657619502367302955014046, −4.19035215073745481549083266540, −3.39706353476294495667323716408, −2.63367624320119422774350126974, −1.31010447287328950626609444765,
1.93368886658693422279271706466, 2.84364404188366636583664930373, 4.10511994525773315682051886405, 4.52905057527839392343462793665, 5.94177054629955112356117586552, 6.98285322945107899792496901225, 7.54822910013709874538695668821, 8.564179704757351928575476757819, 9.179781241190442701080424947827, 9.885866135352904562992878802105