L(s) = 1 | + (0.501 − 0.289i)5-s + 0.198·7-s + 2.48i·11-s + (−4.34 − 2.50i)13-s + (−4.01 + 2.31i)17-s + (3.16 − 2.99i)19-s + (5.14 + 2.97i)23-s + (−2.33 + 4.03i)25-s + (4.51 − 7.82i)29-s + 1.41i·31-s + (0.0995 − 0.0574i)35-s + 11.6i·37-s + (3.60 + 6.25i)41-s + (4.66 + 8.07i)43-s + (−1.86 − 1.07i)47-s + ⋯ |
L(s) = 1 | + (0.224 − 0.129i)5-s + 0.0750·7-s + 0.749i·11-s + (−1.20 − 0.695i)13-s + (−0.973 + 0.562i)17-s + (0.725 − 0.688i)19-s + (1.07 + 0.619i)23-s + (−0.466 + 0.807i)25-s + (0.839 − 1.45i)29-s + 0.253i·31-s + (0.0168 − 0.00971i)35-s + 1.91i·37-s + (0.563 + 0.976i)41-s + (0.710 + 1.23i)43-s + (−0.271 − 0.156i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.289 - 0.957i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.289 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.427678200\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.427678200\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (-3.16 + 2.99i)T \) |
good | 5 | \( 1 + (-0.501 + 0.289i)T + (2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 - 0.198T + 7T^{2} \) |
| 11 | \( 1 - 2.48iT - 11T^{2} \) |
| 13 | \( 1 + (4.34 + 2.50i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (4.01 - 2.31i)T + (8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (-5.14 - 2.97i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-4.51 + 7.82i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 1.41iT - 31T^{2} \) |
| 37 | \( 1 - 11.6iT - 37T^{2} \) |
| 41 | \( 1 + (-3.60 - 6.25i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-4.66 - 8.07i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (1.86 + 1.07i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-1.62 + 2.81i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-4.59 - 7.95i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (2.46 - 4.27i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (3.17 + 1.83i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-8.07 - 13.9i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (5.02 + 8.70i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-1.26 + 0.730i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 0.592iT - 83T^{2} \) |
| 89 | \( 1 + (-2.78 + 4.82i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (9.32 - 5.38i)T + (48.5 - 84.0i)T^{2} \) |
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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
−9.089718233303996133801481006578, −8.131519165013255294781698468891, −7.47833772629738662484992638707, −6.77636818564907906452846438738, −5.89812304918754369625597913766, −4.88599723283550813399125311806, −4.54890392011404308529939536866, −3.14821034922489485802675351492, −2.39794172739609881407907775988, −1.17048001821134030819494353009,
0.49150266241595856459324907850, 2.02045580961731776179915242507, 2.81413259528529090951885551543, 3.89403368945623203479209753896, 4.83045992827666092736317466830, 5.49379592815593599715907882984, 6.50311434180570031846330320167, 7.09786075224983140618093246335, 7.84412885361848948663895821981, 8.916443374642651858024023175915