L(s) = 1 | + (−1 + 1.73i)5-s − 4.56i·7-s − 5.65i·11-s + (3.94 − 2.28i)13-s + (−1 + 1.73i)17-s + (−4 − 1.73i)19-s + (−4.89 + 2.82i)23-s + (0.500 + 0.866i)25-s + (1.89 − 1.09i)29-s − 7.89·31-s + (7.89 + 4.56i)35-s + 1.09i·37-s + (−4.5 − 2.59i)43-s + (7.89 − 4.56i)47-s − 13.7·49-s + ⋯ |
L(s) = 1 | + (−0.447 + 0.774i)5-s − 1.72i·7-s − 1.70i·11-s + (1.09 − 0.632i)13-s + (−0.242 + 0.420i)17-s + (−0.917 − 0.397i)19-s + (−1.02 + 0.589i)23-s + (0.100 + 0.173i)25-s + (0.352 − 0.203i)29-s − 1.41·31-s + (1.33 + 0.770i)35-s + 0.180i·37-s + (−0.686 − 0.396i)43-s + (1.15 − 0.665i)47-s − 1.97·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.977 + 0.211i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.977 + 0.211i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6676593177\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6676593177\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (4 + 1.73i)T \) |
good | 5 | \( 1 + (1 - 1.73i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + 4.56iT - 7T^{2} \) |
| 11 | \( 1 + 5.65iT - 11T^{2} \) |
| 13 | \( 1 + (-3.94 + 2.28i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (1 - 1.73i)T + (-8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (4.89 - 2.82i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.89 + 1.09i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 7.89T + 31T^{2} \) |
| 37 | \( 1 - 1.09iT - 37T^{2} \) |
| 41 | \( 1 + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (4.5 + 2.59i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-7.89 + 4.56i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (10.8 - 6.29i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (5.89 - 10.2i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.94 - 8.57i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-4.39 - 7.61i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-2 + 3.46i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-3.39 + 5.88i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.0505 + 0.0874i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 3.46iT - 83T^{2} \) |
| 89 | \( 1 + (-1.89 + 1.09i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-3.79 - 2.19i)T + (48.5 + 84.0i)T^{2} \) |
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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
−8.310443066701506375592181162299, −7.74771729547796181725441129742, −6.96130919876948927644716162218, −6.27317117144062370610258299569, −5.54357393070144678616009120920, −4.08007133134033340732390577076, −3.74900312200181241021267056200, −2.98171767610243353027852636744, −1.34521071993339748236734242434, −0.21721572510602863287202598633,
1.75523568263743574733326806083, 2.30080454270453657724602016803, 3.72326233336191596975280139638, 4.57911619911788764171015984535, 5.15912975654312876832663793728, 6.18555486886294589240016920299, 6.72547360916542750894872937601, 7.978269197843624607617423961819, 8.393247531561273685018714339192, 9.220246369511728966932876863709