L(s) = 1 | + (0.5 + 0.866i)2-s + (−1.70 − 0.329i)3-s + (−0.499 + 0.866i)4-s + (1.26 + 0.730i)5-s + (−0.565 − 1.63i)6-s + (−2.63 − 0.263i)7-s − 0.999·8-s + (2.78 + 1.11i)9-s + 1.46i·10-s − 3.51·11-s + (1.13 − 1.30i)12-s + (−0.214 − 3.59i)13-s + (−1.08 − 2.41i)14-s + (−1.90 − 1.65i)15-s + (−0.5 − 0.866i)16-s + (−3.79 + 6.57i)17-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.981 − 0.190i)3-s + (−0.249 + 0.433i)4-s + (0.565 + 0.326i)5-s + (−0.230 − 0.668i)6-s + (−0.995 − 0.0995i)7-s − 0.353·8-s + (0.927 + 0.373i)9-s + 0.461i·10-s − 1.06·11-s + (0.327 − 0.377i)12-s + (−0.0595 − 0.998i)13-s + (−0.290 − 0.644i)14-s + (−0.493 − 0.428i)15-s + (−0.125 − 0.216i)16-s + (−0.920 + 1.59i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.763 + 0.645i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.763 + 0.645i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0281182 - 0.0768043i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0281182 - 0.0768043i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 - 0.866i)T \) |
| 3 | \( 1 + (1.70 + 0.329i)T \) |
| 7 | \( 1 + (2.63 + 0.263i)T \) |
| 13 | \( 1 + (0.214 + 3.59i)T \) |
good | 5 | \( 1 + (-1.26 - 0.730i)T + (2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + 3.51T + 11T^{2} \) |
| 17 | \( 1 + (3.79 - 6.57i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + 3.45T + 19T^{2} \) |
| 23 | \( 1 + (-3.12 + 1.80i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (0.170 + 0.0985i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (5.34 + 9.25i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (4.81 - 2.78i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (2.60 + 1.50i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-5.61 - 9.71i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (11.0 + 6.39i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (4.21 - 2.43i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.13 - 0.654i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 - 0.999iT - 61T^{2} \) |
| 67 | \( 1 - 5.52iT - 67T^{2} \) |
| 71 | \( 1 + (-5.33 - 9.24i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-2.94 - 5.09i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.174 + 0.302i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 3.72iT - 83T^{2} \) |
| 89 | \( 1 + (-12.5 + 7.23i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-1.72 - 2.99i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−11.18296099827816763811927266516, −10.44512033276893390084249906379, −9.868755205582722078035634186316, −8.479019884089001443291196612310, −7.52608036364825597029504310236, −6.41438833598445853975747075106, −6.08784856996305123565709538410, −5.10184565784926671873206006971, −3.88457765391448812560470887050, −2.39182029404156342016982220618,
0.04375775456639016189499722381, 1.97913120825842215564371079313, 3.39290188438611830744914001484, 4.79102842250468698566591012312, 5.32425860650155944848675852717, 6.46463546407264316764891469290, 7.15938663220388233091114921492, 9.091387025017799628802470400672, 9.428966668486938900936729839713, 10.47439129640273372082254661465