L(s) = 1 | − 1.73·3-s + (−1.73 − i)4-s + (0.866 + 2.5i)7-s + 2.99·9-s + (2.99 + 1.73i)12-s + (3.46 + i)13-s + (1.99 + 3.46i)16-s + (4.83 + 4.83i)19-s + (−1.49 − 4.33i)21-s + (4.33 − 2.5i)25-s − 5.19·27-s + (1.00 − 5.19i)28-s + (−2.86 + 10.6i)31-s + (−5.19 − 2.99i)36-s + (−11.0 − 2.96i)37-s + ⋯ |
L(s) = 1 | − 1.00·3-s + (−0.866 − 0.5i)4-s + (0.327 + 0.944i)7-s + 0.999·9-s + (0.866 + 0.499i)12-s + (0.960 + 0.277i)13-s + (0.499 + 0.866i)16-s + (1.10 + 1.10i)19-s + (−0.327 − 0.944i)21-s + (0.866 − 0.5i)25-s − 1.00·27-s + (0.188 − 0.981i)28-s + (−0.514 + 1.92i)31-s + (−0.866 − 0.499i)36-s + (−1.81 − 0.487i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.786 - 0.617i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.786 - 0.617i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.749937 + 0.259053i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.749937 + 0.259053i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + 1.73T \) |
| 7 | \( 1 + (-0.866 - 2.5i)T \) |
| 13 | \( 1 + (-3.46 - i)T \) |
good | 2 | \( 1 + (1.73 + i)T^{2} \) |
| 5 | \( 1 + (-4.33 + 2.5i)T^{2} \) |
| 11 | \( 1 + 11iT^{2} \) |
| 17 | \( 1 + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-4.83 - 4.83i)T + 19iT^{2} \) |
| 23 | \( 1 + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (2.86 - 10.6i)T + (-26.8 - 15.5i)T^{2} \) |
| 37 | \( 1 + (11.0 + 2.96i)T + (32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (-35.5 + 20.5i)T^{2} \) |
| 43 | \( 1 + (-9 + 5.19i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (51.0 - 29.5i)T^{2} \) |
| 61 | \( 1 - 15.5T + 61T^{2} \) |
| 67 | \( 1 + (-0.562 - 0.562i)T + 67iT^{2} \) |
| 71 | \( 1 + (61.4 + 35.5i)T^{2} \) |
| 73 | \( 1 + (-2.63 - 0.705i)T + (63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (6.06 - 10.5i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 83iT^{2} \) |
| 89 | \( 1 + (-77.0 - 44.5i)T^{2} \) |
| 97 | \( 1 + (4.42 - 16.5i)T + (-84.0 - 48.5i)T^{2} \) |
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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
−12.11607845841374911950234250385, −10.98544474814420628248582250464, −10.25026962185059283096290603604, −9.161029089044769560269067254618, −8.365634259790819036733879139932, −6.83762142493620906227815171043, −5.64738131824238056118633075569, −5.18031596331125900166035076671, −3.80423190773104554637021019618, −1.39961560219702737429592142104,
0.851910396260609784028907903638, 3.57770513903349666813018849959, 4.60554796710661485156932875889, 5.54543726325998252814720280880, 6.96147991899138037417452249385, 7.77312291924538248562853475793, 9.010362177127423118063092174549, 9.999641889628206096775105757934, 11.00616327066053500289832374302, 11.64996295398045131379231857206