L(s) = 1 | + (−0.659 + 1.14i)2-s + (0.130 + 0.226i)4-s + (1.46 − 1.68i)5-s + (2.12 + 1.57i)7-s − 2.98·8-s + (0.962 + 2.78i)10-s + (2.08 − 1.20i)11-s + 1.69·13-s + (−3.19 + 1.38i)14-s + (1.70 − 2.95i)16-s + (0.831 − 0.480i)17-s + (3.56 + 2.05i)19-s + (0.574 + 0.111i)20-s + 3.17i·22-s + (−2.88 + 4.99i)23-s + ⋯ |
L(s) = 1 | + (−0.466 + 0.807i)2-s + (0.0654 + 0.113i)4-s + (0.655 − 0.755i)5-s + (0.803 + 0.595i)7-s − 1.05·8-s + (0.304 + 0.881i)10-s + (0.629 − 0.363i)11-s + 0.469·13-s + (−0.855 + 0.371i)14-s + (0.425 − 0.737i)16-s + (0.201 − 0.116i)17-s + (0.818 + 0.472i)19-s + (0.128 + 0.0248i)20-s + 0.677i·22-s + (−0.601 + 1.04i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.333 - 0.942i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.333 - 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.05790 + 0.748178i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.05790 + 0.748178i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (-1.46 + 1.68i)T \) |
| 7 | \( 1 + (-2.12 - 1.57i)T \) |
good | 2 | \( 1 + (0.659 - 1.14i)T + (-1 - 1.73i)T^{2} \) |
| 11 | \( 1 + (-2.08 + 1.20i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 1.69T + 13T^{2} \) |
| 17 | \( 1 + (-0.831 + 0.480i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.56 - 2.05i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (2.88 - 4.99i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 5.56iT - 29T^{2} \) |
| 31 | \( 1 + (7.58 - 4.37i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (3.44 + 1.98i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 1.87T + 41T^{2} \) |
| 43 | \( 1 + 10.2iT - 43T^{2} \) |
| 47 | \( 1 + (-8.75 - 5.05i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (6.35 + 11.0i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (3.38 + 5.86i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.98 - 1.14i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-4.15 + 2.39i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 2.24iT - 71T^{2} \) |
| 73 | \( 1 + (7.34 + 12.7i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.892 + 1.54i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 9.62iT - 83T^{2} \) |
| 89 | \( 1 + (0.220 - 0.382i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 12.4T + 97T^{2} \) |
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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
−11.98846105585485299396299751791, −10.95835438171817115127907169473, −9.489712724349381578312820305026, −8.902984802797887233866117277410, −8.156001769264097458907871521263, −7.14098126398046184162348389756, −5.86802735414291785693259728793, −5.29940538477061733795587144402, −3.53198882042815168516875429606, −1.67435019190890270000559975467,
1.36966988218234247756598839420, 2.56898581145317631279649432426, 4.02911821132069680367560410361, 5.63075693898576899942667915750, 6.59198120529284316474434323586, 7.68299402896583412716809893356, 9.013525029100666817038747376820, 9.824475591048940320275623435745, 10.60454323166087569572867823249, 11.25054114669369717347936897916