L(s) = 1 | + (−4.45 + 3.48i)2-s + (7.72 − 31.0i)4-s − 40.4i·5-s + 148. i·7-s + (73.7 + 165. i)8-s + (141. + 180. i)10-s − 608.·11-s + 883.·13-s + (−517. − 661. i)14-s + (−904. − 479. i)16-s − 1.13e3i·17-s − 1.11e3i·19-s + (−1.25e3 − 312. i)20-s + (2.71e3 − 2.11e3i)22-s + 1.34e3·23-s + ⋯ |
L(s) = 1 | + (−0.787 + 0.615i)2-s + (0.241 − 0.970i)4-s − 0.724i·5-s + 1.14i·7-s + (0.407 + 0.913i)8-s + (0.446 + 0.570i)10-s − 1.51·11-s + 1.45·13-s + (−0.705 − 0.902i)14-s + (−0.883 − 0.468i)16-s − 0.951i·17-s − 0.706i·19-s + (−0.702 − 0.174i)20-s + (1.19 − 0.933i)22-s + 0.530·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 + 0.241i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.970 + 0.241i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(1.09605 - 0.134233i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.09605 - 0.134233i\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (4.45 - 3.48i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 40.4iT - 3.12e3T^{2} \) |
| 7 | \( 1 - 148. iT - 1.68e4T^{2} \) |
| 11 | \( 1 + 608.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 883.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.13e3iT - 1.41e6T^{2} \) |
| 19 | \( 1 + 1.11e3iT - 2.47e6T^{2} \) |
| 23 | \( 1 - 1.34e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 1.85e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 - 606. iT - 2.86e7T^{2} \) |
| 37 | \( 1 - 1.36e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 2.12e4iT - 1.15e8T^{2} \) |
| 43 | \( 1 + 1.24e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 1.62e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.34e3iT - 4.18e8T^{2} \) |
| 59 | \( 1 - 3.12e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 6.50e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.35e3iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 2.69e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 5.23e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 3.57e4iT - 3.07e9T^{2} \) |
| 83 | \( 1 - 6.29e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 5.45e4iT - 5.58e9T^{2} \) |
| 97 | \( 1 + 1.42e5T + 8.58e9T^{2} \) |
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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
−12.81510987557134419838726704378, −11.45901546518674664955679600304, −10.47118086079618204545746575021, −9.007183643385948239557574401306, −8.639474264230656341477987578605, −7.32989632194224092581430726359, −5.80382433758882510689025738054, −5.03715617007176707004992508438, −2.48737282816977234546048265287, −0.68096574735287298065108836409,
1.08120847582640209705496781823, 2.86692095914772089440202392993, 4.08369416305890113663602020835, 6.28253445427515889136558011437, 7.57367760532358049501948381616, 8.330875306765483024165421654173, 9.951086603495229514960086413366, 10.69738152248627283619207968073, 11.21840676646599226555988204483, 12.92874384679686586060003148178