| L(s) = 1 | + (607. − 1.05e3i)5-s + (4.03e3 − 2.32e3i)7-s + (3.67e3 − 2.12e3i)11-s + (−1.39e4 + 2.40e4i)13-s + 1.91e3·17-s − 9.66e4i·19-s + (2.83e5 + 1.63e5i)23-s + (−5.42e5 − 9.40e5i)25-s + (3.99e5 + 6.91e5i)29-s + (5.61e5 + 3.24e5i)31-s − 5.65e6i·35-s + 2.23e6·37-s + (4.91e5 − 8.51e5i)41-s + (3.68e6 − 2.12e6i)43-s + (2.98e6 − 1.72e6i)47-s + ⋯ |
| L(s) = 1 | + (0.971 − 1.68i)5-s + (1.67 − 0.969i)7-s + (0.250 − 0.144i)11-s + (−0.486 + 0.843i)13-s + 0.0229·17-s − 0.741i·19-s + (1.01 + 0.585i)23-s + (−1.38 − 2.40i)25-s + (0.564 + 0.977i)29-s + (0.607 + 0.351i)31-s − 3.76i·35-s + 1.19·37-s + (0.173 − 0.301i)41-s + (1.07 − 0.622i)43-s + (0.611 − 0.353i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.206 + 0.978i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.206 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(4.078718059\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.078718059\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (-607. + 1.05e3i)T + (-1.95e5 - 3.38e5i)T^{2} \) |
| 7 | \( 1 + (-4.03e3 + 2.32e3i)T + (2.88e6 - 4.99e6i)T^{2} \) |
| 11 | \( 1 + (-3.67e3 + 2.12e3i)T + (1.07e8 - 1.85e8i)T^{2} \) |
| 13 | \( 1 + (1.39e4 - 2.40e4i)T + (-4.07e8 - 7.06e8i)T^{2} \) |
| 17 | \( 1 - 1.91e3T + 6.97e9T^{2} \) |
| 19 | \( 1 + 9.66e4iT - 1.69e10T^{2} \) |
| 23 | \( 1 + (-2.83e5 - 1.63e5i)T + (3.91e10 + 6.78e10i)T^{2} \) |
| 29 | \( 1 + (-3.99e5 - 6.91e5i)T + (-2.50e11 + 4.33e11i)T^{2} \) |
| 31 | \( 1 + (-5.61e5 - 3.24e5i)T + (4.26e11 + 7.38e11i)T^{2} \) |
| 37 | \( 1 - 2.23e6T + 3.51e12T^{2} \) |
| 41 | \( 1 + (-4.91e5 + 8.51e5i)T + (-3.99e12 - 6.91e12i)T^{2} \) |
| 43 | \( 1 + (-3.68e6 + 2.12e6i)T + (5.84e12 - 1.01e13i)T^{2} \) |
| 47 | \( 1 + (-2.98e6 + 1.72e6i)T + (1.19e13 - 2.06e13i)T^{2} \) |
| 53 | \( 1 - 5.99e6T + 6.22e13T^{2} \) |
| 59 | \( 1 + (1.21e7 + 6.99e6i)T + (7.34e13 + 1.27e14i)T^{2} \) |
| 61 | \( 1 + (6.50e5 + 1.12e6i)T + (-9.58e13 + 1.66e14i)T^{2} \) |
| 67 | \( 1 + (9.82e6 + 5.67e6i)T + (2.03e14 + 3.51e14i)T^{2} \) |
| 71 | \( 1 - 6.82e6iT - 6.45e14T^{2} \) |
| 73 | \( 1 - 3.84e7T + 8.06e14T^{2} \) |
| 79 | \( 1 + (-1.73e6 + 1.00e6i)T + (7.58e14 - 1.31e15i)T^{2} \) |
| 83 | \( 1 + (3.73e7 - 2.15e7i)T + (1.12e15 - 1.95e15i)T^{2} \) |
| 89 | \( 1 + 7.34e7T + 3.93e15T^{2} \) |
| 97 | \( 1 + (1.67e7 + 2.90e7i)T + (-3.91e15 + 6.78e15i)T^{2} \) |
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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
−9.319569486105540282971588696680, −8.797885313177149139026622267340, −7.87832143224032936814850346001, −6.86092228572891667475996935525, −5.44875394396695207386316704293, −4.79887911958517455128523893865, −4.22708170526852717549724964928, −2.23247468807319511368017812179, −1.26932725287293806824106634408, −0.826772601200704497146499391243,
1.24481387825610452253355698510, 2.38615046746673536817604176275, 2.77979304106526919410646390305, 4.45977730452019175440456981782, 5.60564850429418748762710514566, 6.18071706429193296618763697026, 7.40441950393468608907803482594, 8.100869211177656512861559771406, 9.294486449855983027506840948801, 10.19338210238040279907059769233
