| L(s) = 1 | + 2i·3-s + (−1 + 2i)5-s + 2i·7-s − 9-s − 11-s + (−4 − 2i)15-s − 4i·17-s − 4·19-s − 4·21-s + 6i·23-s + (−3 − 4i)25-s + 4i·27-s − 2·29-s + 8·31-s − 2i·33-s + ⋯ |
| L(s) = 1 | + 1.15i·3-s + (−0.447 + 0.894i)5-s + 0.755i·7-s − 0.333·9-s − 0.301·11-s + (−1.03 − 0.516i)15-s − 0.970i·17-s − 0.917·19-s − 0.872·21-s + 1.25i·23-s + (−0.600 − 0.800i)25-s + 0.769i·27-s − 0.371·29-s + 1.43·31-s − 0.348i·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.247813 + 1.04975i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.247813 + 1.04975i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (1 - 2i)T \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 - 2iT - 3T^{2} \) |
| 7 | \( 1 - 2iT - 7T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 + 4iT - 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 - 6iT - 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 + 4iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 - 6iT - 43T^{2} \) |
| 47 | \( 1 + 2iT - 47T^{2} \) |
| 53 | \( 1 - 12iT - 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 - 14T + 61T^{2} \) |
| 67 | \( 1 - 10iT - 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 + 4iT - 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + 2iT - 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 + 8iT - 97T^{2} \) |
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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
−11.36572114004635848600956117352, −10.55787383037823770628685891191, −9.810647581571463515380527240658, −9.003076710520285613371303604748, −7.907089846034882614817261646092, −6.87215146085167049520911596624, −5.69791440170000227383143639971, −4.65709912619538378777439344959, −3.60347317798932890194676622399, −2.54521012174282195832422725661,
0.68484837455258599651846059410, 2.03082661722900013689147938860, 3.86596933353002954570529155011, 4.85343387815503816826307657787, 6.28286464040817646115025342283, 7.02631659470772875607280564961, 8.169014449639384430285692889352, 8.418010201702661654127423564143, 9.930172879529839923082036369264, 10.78848901929084714073547113363
