| L(s) = 1 | + 1.02e5i·3-s − 9.85e8i·7-s − 1.04e10·9-s + 3.70e11·13-s + 3.99e13i·19-s + 1.00e14·21-s + 4.76e14·25-s − 1.06e15i·27-s + 1.25e15i·31-s − 5.77e16·37-s + 3.78e16i·39-s − 9.98e16i·43-s − 4.12e17·49-s − 4.08e18·57-s − 1.08e19·61-s + ⋯ |
| L(s) = 1 | + 0.999i·3-s − 1.31i·7-s − 0.999·9-s + 0.744·13-s + 1.49i·19-s + 1.31·21-s + 0.999·25-s − 0.999i·27-s + 0.275i·31-s − 1.97·37-s + 0.744i·39-s − 0.704i·43-s − 0.738·49-s − 1.49·57-s − 1.94·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(11)\) |
\(\approx\) |
\(0.9790499059\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9790499059\) |
| \(L(\frac{23}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 1.02e5iT \) |
| good | 5 | \( 1 - 4.76e14T^{2} \) |
| 7 | \( 1 + 9.85e8iT - 5.58e17T^{2} \) |
| 11 | \( 1 + 7.40e21T^{2} \) |
| 13 | \( 1 - 3.70e11T + 2.47e23T^{2} \) |
| 17 | \( 1 - 6.90e25T^{2} \) |
| 19 | \( 1 - 3.99e13iT - 7.14e26T^{2} \) |
| 23 | \( 1 + 3.94e28T^{2} \) |
| 29 | \( 1 - 5.13e30T^{2} \) |
| 31 | \( 1 - 1.25e15iT - 2.08e31T^{2} \) |
| 37 | \( 1 + 5.77e16T + 8.55e32T^{2} \) |
| 41 | \( 1 - 7.38e33T^{2} \) |
| 43 | \( 1 + 9.98e16iT - 2.00e34T^{2} \) |
| 47 | \( 1 + 1.30e35T^{2} \) |
| 53 | \( 1 - 1.62e36T^{2} \) |
| 59 | \( 1 + 1.54e37T^{2} \) |
| 61 | \( 1 + 1.08e19T + 3.10e37T^{2} \) |
| 67 | \( 1 + 2.90e19iT - 2.22e38T^{2} \) |
| 71 | \( 1 + 7.52e38T^{2} \) |
| 73 | \( 1 - 3.90e19T + 1.34e39T^{2} \) |
| 79 | \( 1 + 9.04e18iT - 7.08e39T^{2} \) |
| 83 | \( 1 + 1.99e40T^{2} \) |
| 89 | \( 1 - 8.65e40T^{2} \) |
| 97 | \( 1 - 1.13e21T + 5.27e41T^{2} \) |
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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
−10.74326162380519472862600068202, −10.36092032577083484051093571382, −9.030753485176503830026704288289, −7.88150414966449203786005979805, −6.49365336685860091768027342210, −5.16901766777271236725650525687, −3.99938046599355793178196937154, −3.31234709276687591973273586388, −1.49525711875435765668768206704, −0.21015745495030663549511318643,
1.06811041151747869741233765044, 2.22340802475867660586197528011, 3.12367754584581262628475323013, 5.03618659842459905519228988672, 6.07891838364911171000807752338, 7.06504328494644327583825280695, 8.465355261345552044831571214477, 9.088785234460964223215790014589, 10.93619281492319716921710891732, 11.89202840074370666272382857989
