| L(s) = 1 | + (−0.984 + 0.173i)2-s + (−1.03 + 1.39i)3-s + (0.939 − 0.342i)4-s + (−1.21 + 1.87i)5-s + (0.775 − 1.54i)6-s + (−1.48 − 0.856i)7-s + (−0.866 + 0.5i)8-s + (−0.867 − 2.87i)9-s + (0.873 − 2.05i)10-s + (0.232 − 0.134i)11-s + (−0.494 + 1.65i)12-s + (−2.84 − 2.38i)13-s + (1.61 + 0.586i)14-s + (−1.34 − 3.63i)15-s + (0.766 − 0.642i)16-s + (0.267 + 1.51i)17-s + ⋯ |
| L(s) = 1 | + (−0.696 + 0.122i)2-s + (−0.596 + 0.802i)3-s + (0.469 − 0.171i)4-s + (−0.544 + 0.838i)5-s + (0.316 − 0.632i)6-s + (−0.560 − 0.323i)7-s + (−0.306 + 0.176i)8-s + (−0.289 − 0.957i)9-s + (0.276 − 0.650i)10-s + (0.0702 − 0.0405i)11-s + (−0.142 + 0.479i)12-s + (−0.787 − 0.661i)13-s + (0.430 + 0.156i)14-s + (−0.348 − 0.937i)15-s + (0.191 − 0.160i)16-s + (0.0648 + 0.367i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.625 + 0.780i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.625 + 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.335172 - 0.160995i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.335172 - 0.160995i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.984 - 0.173i)T \) |
| 3 | \( 1 + (1.03 - 1.39i)T \) |
| 5 | \( 1 + (1.21 - 1.87i)T \) |
| 19 | \( 1 + (-2.37 + 3.65i)T \) |
| good | 7 | \( 1 + (1.48 + 0.856i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.232 + 0.134i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (2.84 + 2.38i)T + (2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-0.267 - 1.51i)T + (-15.9 + 5.81i)T^{2} \) |
| 23 | \( 1 + (0.801 - 0.291i)T + (17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (0.356 - 2.02i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (-0.244 - 0.141i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 9.38T + 37T^{2} \) |
| 41 | \( 1 + (1.94 - 1.62i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (-2.01 + 5.53i)T + (-32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (-1.42 + 8.10i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + (-1.09 - 2.99i)T + (-40.6 + 34.0i)T^{2} \) |
| 59 | \( 1 + (2.44 + 13.8i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (1.11 - 0.404i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (0.393 - 2.23i)T + (-62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (-2.24 - 0.817i)T + (54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + (4.67 + 5.57i)T + (-12.6 + 71.8i)T^{2} \) |
| 79 | \( 1 + (-3.38 - 4.03i)T + (-13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-7.43 + 12.8i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (10.6 + 8.94i)T + (15.4 + 87.6i)T^{2} \) |
| 97 | \( 1 + (-1.04 - 5.92i)T + (-91.1 + 33.1i)T^{2} \) |
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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
−10.43236244245189894933687825915, −9.956634680193104381236455675424, −9.077744447716357923703197498507, −7.88083962898048193756519955732, −7.01551235943730187960966716278, −6.23508167295092094118712481657, −5.09158771561619968728455616388, −3.78841910149131469471098087077, −2.81915376546087165832953078654, −0.31891562458110003454281507701,
1.20915245983383013353237972431, 2.64184312630548067077455497186, 4.31845629666922519266476152185, 5.51804227354836919841156038640, 6.45619325516280469227666102643, 7.50909386935203964676610939222, 8.032307728534646850998365617704, 9.186874084148107255378674180201, 9.796795056213182207075686398114, 11.03914092277117626203916788354
