L(s) = 1 | + (1.02 + 0.589i)2-s + (−0.305 − 0.529i)4-s + (−2.16 − 3.75i)5-s − 3.07i·8-s − 5.10i·10-s + (1.87 + 1.08i)11-s + (−2.25 + 1.30i)13-s + (1.20 − 2.08i)16-s − 1.17·17-s − 2.41i·19-s + (−1.32 + 2.29i)20-s + (1.27 + 2.20i)22-s + (3.16 − 1.82i)23-s + (−6.88 + 11.9i)25-s − 3.06·26-s + ⋯ |
L(s) = 1 | + (0.721 + 0.416i)2-s + (−0.152 − 0.264i)4-s + (−0.968 − 1.67i)5-s − 1.08i·8-s − 1.61i·10-s + (0.564 + 0.325i)11-s + (−0.624 + 0.360i)13-s + (0.300 − 0.520i)16-s − 0.284·17-s − 0.554i·19-s + (−0.296 + 0.513i)20-s + (0.271 + 0.470i)22-s + (0.659 − 0.380i)23-s + (−1.37 + 2.38i)25-s − 0.601·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.958 + 0.285i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.958 + 0.285i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9532381007\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9532381007\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (-1.02 - 0.589i)T + (1 + 1.73i)T^{2} \) |
| 5 | \( 1 + (2.16 + 3.75i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-1.87 - 1.08i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (2.25 - 1.30i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + 1.17T + 17T^{2} \) |
| 19 | \( 1 + 2.41iT - 19T^{2} \) |
| 23 | \( 1 + (-3.16 + 1.82i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (0.589 + 0.340i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (5.67 - 3.27i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 5.10T + 37T^{2} \) |
| 41 | \( 1 + (3.68 + 6.38i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (2.12 - 3.68i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-3.57 + 6.18i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 3.23iT - 53T^{2} \) |
| 59 | \( 1 + (-2.91 - 5.05i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (6.21 + 3.58i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (3.32 + 5.76i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 1.95iT - 71T^{2} \) |
| 73 | \( 1 - 11.9iT - 73T^{2} \) |
| 79 | \( 1 + (-4.87 + 8.44i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-0.796 + 1.37i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 6.09T + 89T^{2} \) |
| 97 | \( 1 + (-2.36 - 1.36i)T + (48.5 + 84.0i)T^{2} \) |
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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
−9.007878277842606479832476495294, −8.701794371690907589093979142096, −7.41395098244743926956815677379, −6.86996636889090799536393837427, −5.59217350136514908072172163615, −4.89633827468405833389032384019, −4.39247286917420281663390750765, −3.55911753092274985420268297600, −1.57961353441798525429258202958, −0.31162978078402455326804299742,
2.24581647044664219191249732840, 3.26586160228228403071986877552, 3.65085861547326451575902733527, 4.65215426232882179061919360765, 5.80168986423399470498794080489, 6.79849092315296368821064552460, 7.52395647877147409778039279267, 8.159148681372757562602977176655, 9.206220883691863136274107634335, 10.32561380868078481372959184087