| L(s) = 1 | + (121.5 + 210. i)3-s + (4.98e3 − 8.63e3i)5-s + (6.34e3 − 4.40e4i)7-s + (−2.95e4 + 5.11e4i)9-s + (4.23e5 + 7.33e5i)11-s − 7.79e5·13-s + 2.42e6·15-s + (5.75e6 + 9.96e6i)17-s + (−7.03e6 + 1.21e7i)19-s + (1.00e7 − 4.01e6i)21-s + (2.60e7 − 4.50e7i)23-s + (−2.53e7 − 4.38e7i)25-s − 1.43e7·27-s + 1.03e8·29-s + (−2.99e7 − 5.18e7i)31-s + ⋯ |
| L(s) = 1 | + (0.288 + 0.499i)3-s + (0.713 − 1.23i)5-s + (0.142 − 0.989i)7-s + (−0.166 + 0.288i)9-s + (0.793 + 1.37i)11-s − 0.582·13-s + 0.824·15-s + (0.982 + 1.70i)17-s + (−0.652 + 1.12i)19-s + (0.536 − 0.214i)21-s + (0.843 − 1.46i)23-s + (−0.518 − 0.898i)25-s − 0.192·27-s + 0.934·29-s + (−0.187 − 0.325i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0796i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.996 + 0.0796i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(3.03116 - 0.120860i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.03116 - 0.120860i\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-121.5 - 210. i)T \) |
| 7 | \( 1 + (-6.34e3 + 4.40e4i)T \) |
| good | 5 | \( 1 + (-4.98e3 + 8.63e3i)T + (-2.44e7 - 4.22e7i)T^{2} \) |
| 11 | \( 1 + (-4.23e5 - 7.33e5i)T + (-1.42e11 + 2.47e11i)T^{2} \) |
| 13 | \( 1 + 7.79e5T + 1.79e12T^{2} \) |
| 17 | \( 1 + (-5.75e6 - 9.96e6i)T + (-1.71e13 + 2.96e13i)T^{2} \) |
| 19 | \( 1 + (7.03e6 - 1.21e7i)T + (-5.82e13 - 1.00e14i)T^{2} \) |
| 23 | \( 1 + (-2.60e7 + 4.50e7i)T + (-4.76e14 - 8.25e14i)T^{2} \) |
| 29 | \( 1 - 1.03e8T + 1.22e16T^{2} \) |
| 31 | \( 1 + (2.99e7 + 5.18e7i)T + (-1.27e16 + 2.20e16i)T^{2} \) |
| 37 | \( 1 + (-2.06e8 + 3.57e8i)T + (-8.89e16 - 1.54e17i)T^{2} \) |
| 41 | \( 1 - 8.38e8T + 5.50e17T^{2} \) |
| 43 | \( 1 - 7.99e8T + 9.29e17T^{2} \) |
| 47 | \( 1 + (2.61e8 - 4.52e8i)T + (-1.23e18 - 2.14e18i)T^{2} \) |
| 53 | \( 1 + (-7.45e8 - 1.29e9i)T + (-4.63e18 + 8.02e18i)T^{2} \) |
| 59 | \( 1 + (-1.94e9 - 3.37e9i)T + (-1.50e19 + 2.61e19i)T^{2} \) |
| 61 | \( 1 + (-4.09e9 + 7.10e9i)T + (-2.17e19 - 3.76e19i)T^{2} \) |
| 67 | \( 1 + (2.66e9 + 4.60e9i)T + (-6.10e19 + 1.05e20i)T^{2} \) |
| 71 | \( 1 + 5.14e9T + 2.31e20T^{2} \) |
| 73 | \( 1 + (7.07e9 + 1.22e10i)T + (-1.56e20 + 2.71e20i)T^{2} \) |
| 79 | \( 1 + (3.09e9 - 5.35e9i)T + (-3.73e20 - 6.47e20i)T^{2} \) |
| 83 | \( 1 - 8.06e9T + 1.28e21T^{2} \) |
| 89 | \( 1 + (-3.41e10 + 5.91e10i)T + (-1.38e21 - 2.40e21i)T^{2} \) |
| 97 | \( 1 - 1.16e11T + 7.15e21T^{2} \) |
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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
−12.44561727020785511142282967760, −10.52286618101764879030536995150, −9.866660662508858341863210303291, −8.819436449620929747985356516817, −7.68148275160180846558651817151, −6.13958532940516579659753861346, −4.68747205523819744642302339788, −4.03511452272227780715315470177, −1.97801203018024465842884105317, −0.978711925869549678823631075438,
0.941647236502738909937926852604, 2.54928247642773050663368440873, 3.08363699698864640148219901869, 5.34224083491583512132700387440, 6.38690211324556971307653200829, 7.33893278075055864982021422316, 8.834927026731283785292432260822, 9.675661489991448139242991483392, 11.18148427168537743295979194970, 11.83435716132110659288527917787
