L(s) = 1 | + (−0.866 − 0.5i)2-s + (0.499 + 0.866i)4-s − i·5-s + (2.59 − 1.5i)7-s − 0.999i·8-s + (−0.5 + 0.866i)10-s + (−0.232 − 0.133i)11-s + (0.866 + 3.5i)13-s − 3·14-s + (−0.5 + 0.866i)16-s + (2 + 3.46i)17-s + (4.96 − 2.86i)19-s + (0.866 − 0.499i)20-s + (0.133 + 0.232i)22-s + (1.73 − 3i)23-s + ⋯ |
L(s) = 1 | + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s − 0.447i·5-s + (0.981 − 0.566i)7-s − 0.353i·8-s + (−0.158 + 0.273i)10-s + (−0.0699 − 0.0403i)11-s + (0.240 + 0.970i)13-s − 0.801·14-s + (−0.125 + 0.216i)16-s + (0.485 + 0.840i)17-s + (1.13 − 0.657i)19-s + (0.193 − 0.111i)20-s + (0.0285 + 0.0494i)22-s + (0.361 − 0.625i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.702 + 0.711i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.702 + 0.711i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.420830827\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.420830827\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 + 0.5i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + iT \) |
| 13 | \( 1 + (-0.866 - 3.5i)T \) |
good | 7 | \( 1 + (-2.59 + 1.5i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (0.232 + 0.133i)T + (5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (-2 - 3.46i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-4.96 + 2.86i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.73 + 3i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.732 + 1.26i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 4.92iT - 31T^{2} \) |
| 37 | \( 1 + (5.13 + 2.96i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-3.46 - 2i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (3 + 5.19i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 6.46iT - 47T^{2} \) |
| 53 | \( 1 - 0.267T + 53T^{2} \) |
| 59 | \( 1 + (-9.92 + 5.73i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.267 - 0.464i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.26 - 0.732i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-11.1 + 6.46i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + 6.92iT - 73T^{2} \) |
| 79 | \( 1 - 3.07T + 79T^{2} \) |
| 83 | \( 1 + 9.46iT - 83T^{2} \) |
| 89 | \( 1 + (-12.2 - 7.06i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-7.26 + 4.19i)T + (48.5 - 84.0i)T^{2} \) |
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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
−9.609141117060972563105060950752, −8.858667869542109779709892581906, −8.167411013090395168725179062426, −7.40216830499976534252252097908, −6.54924859775625570488700779780, −5.26013810330017264984537899787, −4.43760280876518864695344612559, −3.42349608106645822069929463647, −1.95699256349154985658221363673, −0.989877565214531816736208764081,
1.13197727366347250905797577908, 2.46473093580853033710750156575, 3.56620769204012685251646693123, 5.22028760132195090630312365516, 5.48215738354254951062616620048, 6.71098240277930954339833224241, 7.67503664069155845111921605553, 8.050525675847891896833512009260, 9.026709687332962814081391364541, 9.862896833096185143461726991375