L(s) = 1 | + (−1.5 − 2.59i)5-s + (−2 + 3.46i)7-s + (0.5 + 0.866i)13-s + 3·17-s + 4·19-s + (−2 + 3.46i)25-s + (4.5 − 7.79i)29-s + (−2 − 3.46i)31-s + 12·35-s − 37-s + (3 + 5.19i)41-s + (4 − 6.92i)43-s + (6 − 10.3i)47-s + (−4.49 − 7.79i)49-s + 6·53-s + ⋯ |
L(s) = 1 | + (−0.670 − 1.16i)5-s + (−0.755 + 1.30i)7-s + (0.138 + 0.240i)13-s + 0.727·17-s + 0.917·19-s + (−0.400 + 0.692i)25-s + (0.835 − 1.44i)29-s + (−0.359 − 0.622i)31-s + 2.02·35-s − 0.164·37-s + (0.468 + 0.811i)41-s + (0.609 − 1.05i)43-s + (0.875 − 1.51i)47-s + (−0.642 − 1.11i)49-s + 0.824·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.268855031\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.268855031\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (1.5 + 2.59i)T + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (2 - 3.46i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.5 - 0.866i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 - 3T + 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-4.5 + 7.79i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (2 + 3.46i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + T + 37T^{2} \) |
| 41 | \( 1 + (-3 - 5.19i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-4 + 6.92i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-6 + 10.3i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.5 + 0.866i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (2 + 3.46i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 - 11T + 73T^{2} \) |
| 79 | \( 1 + (8 - 13.8i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-6 + 10.3i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 3T + 89T^{2} \) |
| 97 | \( 1 + (1 - 1.73i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.425186132969605128752203026029, −8.766467431775290903638156691973, −8.137259291917816043118933199717, −7.22031512434760449912064748604, −5.99649696803467346375902529447, −5.45783566444143819342691536576, −4.44449747710178832331294550631, −3.45603402797998639510096728836, −2.30852222462586609344772542945, −0.70479586560519622839123601216,
0.995591272542316057194106228394, 3.00039605802937088640114894477, 3.42929634736446864710308860066, 4.39709510210782604752216641986, 5.69100725606541307670848786556, 6.72467639090079128148832262360, 7.28185181323512946689732896319, 7.77398447808587926281510681957, 9.013913198769548838075093542767, 9.971464619570892566025350264745