| L(s) = 1 | + 2.44·5-s − 2.44·7-s + (−3 − 1.41i)11-s + 4.24i·13-s + 3.46i·17-s + 1.41i·23-s + 0.999·25-s + 6.92i·29-s + 3.46i·31-s − 5.99·35-s − 2·37-s + 6.92i·41-s + 9.79·43-s + 7.07i·47-s − 1.00·49-s + ⋯ |
| L(s) = 1 | + 1.09·5-s − 0.925·7-s + (−0.904 − 0.426i)11-s + 1.17i·13-s + 0.840i·17-s + 0.294i·23-s + 0.199·25-s + 1.28i·29-s + 0.622i·31-s − 1.01·35-s − 0.328·37-s + 1.08i·41-s + 1.49·43-s + 1.03i·47-s − 0.142·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0829 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1584 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0829 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.292649141\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.292649141\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 + (3 + 1.41i)T \) |
| good | 5 | \( 1 - 2.44T + 5T^{2} \) |
| 7 | \( 1 + 2.44T + 7T^{2} \) |
| 13 | \( 1 - 4.24iT - 13T^{2} \) |
| 17 | \( 1 - 3.46iT - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 1.41iT - 23T^{2} \) |
| 29 | \( 1 - 6.92iT - 29T^{2} \) |
| 31 | \( 1 - 3.46iT - 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 - 6.92iT - 41T^{2} \) |
| 43 | \( 1 - 9.79T + 43T^{2} \) |
| 47 | \( 1 - 7.07iT - 47T^{2} \) |
| 53 | \( 1 - 7.34T + 53T^{2} \) |
| 59 | \( 1 - 14.1iT - 59T^{2} \) |
| 61 | \( 1 + 12.7iT - 61T^{2} \) |
| 67 | \( 1 + 3.46iT - 67T^{2} \) |
| 71 | \( 1 + 9.89iT - 71T^{2} \) |
| 73 | \( 1 + 8.48iT - 73T^{2} \) |
| 79 | \( 1 - 2.44T + 79T^{2} \) |
| 83 | \( 1 + 12T + 83T^{2} \) |
| 89 | \( 1 - 4.89T + 89T^{2} \) |
| 97 | \( 1 + 8T + 97T^{2} \) |
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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
−9.462969003793143193583351689056, −9.127098027801626279611844521286, −8.103044883804338419811682447651, −7.06254890196777251451630837519, −6.28507150974543660232269057913, −5.73230646352994958613664358022, −4.73901129700221629238549793581, −3.54664289994453280483871820388, −2.59159959974899513571196271239, −1.51207420712644413207972932992,
0.47832747922043475298689053588, 2.26280577542139297460551830189, 2.86482435180074017453468120525, 4.13030854861657757893314458726, 5.45598037436489630040522511266, 5.70324226126238849653780185111, 6.77903101710874446582898571255, 7.54367925778246276666291459829, 8.464328648530564242515149789640, 9.456487389225845827531166337589
