L(s) = 1 | + (1 − 2i)5-s − i·7-s + 2·11-s + 2i·13-s + 2·19-s + 8i·23-s + (−3 − 4i)25-s + 2·29-s + 6·31-s + (−2 − i)35-s − 8i·37-s + 10·41-s + 12i·47-s − 49-s + 2i·53-s + ⋯ |
L(s) = 1 | + (0.447 − 0.894i)5-s − 0.377i·7-s + 0.603·11-s + 0.554i·13-s + 0.458·19-s + 1.66i·23-s + (−0.600 − 0.800i)25-s + 0.371·29-s + 1.07·31-s + (−0.338 − 0.169i)35-s − 1.31i·37-s + 1.56·41-s + 1.75i·47-s − 0.142·49-s + 0.274i·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.342273358\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.342273358\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-1 + 2i)T \) |
| 7 | \( 1 + iT \) |
good | 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 - 8iT - 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 - 6T + 31T^{2} \) |
| 37 | \( 1 + 8iT - 37T^{2} \) |
| 41 | \( 1 - 10T + 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 - 12iT - 47T^{2} \) |
| 53 | \( 1 - 2iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 - 4iT - 67T^{2} \) |
| 71 | \( 1 - 14T + 71T^{2} \) |
| 73 | \( 1 + 2iT - 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 - 16iT - 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 + 2iT - 97T^{2} \) |
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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
−8.146996694777704946128525018144, −7.54626143581179443777962769437, −6.72032754551153958501931044496, −5.93448955554269360501734963314, −5.32135659856487585816291865948, −4.39505619807262416827813399651, −3.89628260787520350511351381059, −2.72909928330320160194659919123, −1.62483642983660553365881613933, −0.881645498449283600742640052979,
0.867766105126444131128988026296, 2.18139691633490299136085795014, 2.83111565149064657528697172348, 3.65484075242296012289333356383, 4.65504719002921205025060855342, 5.44214532972921304361593836805, 6.41333172168074011557769800479, 6.55009333232478299097879497822, 7.56493494561611626219427935582, 8.295129513648703418563728464478