L(s) = 1 | + (−2 − 1.73i)7-s − 7·13-s + (−4 + 6.92i)19-s + (2.5 + 4.33i)25-s + (−5.5 − 9.52i)31-s + (0.5 − 0.866i)37-s − 13·43-s + (1.00 + 6.92i)49-s + (0.5 − 0.866i)61-s + (−5.5 − 9.52i)67-s + (5 + 8.66i)73-s + (6.5 − 11.2i)79-s + (14 + 12.1i)91-s − 19·97-s + (3.5 − 6.06i)103-s + ⋯ |
L(s) = 1 | + (−0.755 − 0.654i)7-s − 1.94·13-s + (−0.917 + 1.58i)19-s + (0.5 + 0.866i)25-s + (−0.987 − 1.71i)31-s + (0.0821 − 0.142i)37-s − 1.98·43-s + (0.142 + 0.989i)49-s + (0.0640 − 0.110i)61-s + (−0.671 − 1.16i)67-s + (0.585 + 1.01i)73-s + (0.731 − 1.26i)79-s + (1.46 + 1.27i)91-s − 1.92·97-s + (0.344 − 0.597i)103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (2 + 1.73i)T \) |
good | 5 | \( 1 + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 7T + 13T^{2} \) |
| 17 | \( 1 + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (4 - 6.92i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + (5.5 + 9.52i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.5 + 0.866i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 13T + 43T^{2} \) |
| 47 | \( 1 + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.5 + 0.866i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (5.5 + 9.52i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + (-5 - 8.66i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-6.5 + 11.2i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 19T + 97T^{2} \) |
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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
−9.904687386031125116896321677541, −9.291735066432182844123465416347, −8.001563899498960574820874494160, −7.32659321742139427561074640349, −6.46392234880406541068158958085, −5.41228734321090273550459107508, −4.32295978299945350176159014135, −3.33131925600993062102943193966, −2.01116051270762943663652030735, 0,
2.26555039051722170913284208592, 3.09513733736063924369718113462, 4.62216698753173184103127266406, 5.30836994320640864202856382895, 6.64877547330098715349704762108, 7.06476342203788797954215674069, 8.381092038084311284602853670839, 9.120512558456089903221012837588, 9.887322470137963680809190392534