L(s) = 1 | + (3.34 − 1.93i)5-s + (−3.23 − 1.86i)7-s + (0.517 − 0.896i)11-s + (−2.23 − 3.86i)13-s − 1.79i·17-s − 1.73i·19-s + (4.38 + 7.58i)23-s + (4.96 − 8.59i)25-s + (−6.69 − 3.86i)29-s + (−6.46 + 3.73i)31-s − 14.4·35-s + 0.464·37-s + (6.69 − 3.86i)41-s + (0.464 + 0.267i)43-s + (−2.31 + 4.00i)47-s + ⋯ |
L(s) = 1 | + (1.49 − 0.863i)5-s + (−1.22 − 0.705i)7-s + (0.156 − 0.270i)11-s + (−0.619 − 1.07i)13-s − 0.434i·17-s − 0.397i·19-s + (0.913 + 1.58i)23-s + (0.992 − 1.71i)25-s + (−1.24 − 0.717i)29-s + (−1.16 + 0.670i)31-s − 2.43·35-s + 0.0762·37-s + (1.04 − 0.603i)41-s + (0.0707 + 0.0408i)43-s + (−0.337 + 0.583i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.819 + 0.573i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.819 + 0.573i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.424217703\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.424217703\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-3.34 + 1.93i)T + (2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (3.23 + 1.86i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.517 + 0.896i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (2.23 + 3.86i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + 1.79iT - 17T^{2} \) |
| 19 | \( 1 + 1.73iT - 19T^{2} \) |
| 23 | \( 1 + (-4.38 - 7.58i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (6.69 + 3.86i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (6.46 - 3.73i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 0.464T + 37T^{2} \) |
| 41 | \( 1 + (-6.69 + 3.86i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.464 - 0.267i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (2.31 - 4.00i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + 3.58iT - 53T^{2} \) |
| 59 | \( 1 + (6.17 + 10.6i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (5.69 - 9.86i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-5.42 + 3.13i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 11.3T + 71T^{2} \) |
| 73 | \( 1 + 3.92T + 73T^{2} \) |
| 79 | \( 1 + (4.16 + 2.40i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-1.03 + 1.79i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 1.79iT - 89T^{2} \) |
| 97 | \( 1 + (3.5 - 6.06i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−8.964039085575768450282137849545, −7.64991527721019172643578187125, −7.08624884571755496167970366145, −6.04616171405843982326523812007, −5.56969119082098226436653056596, −4.83049841654072543309876653002, −3.58415818743394112554873202381, −2.78472338743213677801880109541, −1.56711439493954975518367415386, −0.43115137269239682622422149310,
1.78950819188395781485461708774, 2.49103777139732255261172446623, 3.26773249999508193622474168558, 4.47083527278406768476409178661, 5.63582350532599118533710198799, 6.10892475152240024541516995193, 6.77868161869313004129007442586, 7.33137680256421391200382373963, 8.806992429726200180648876341566, 9.345148090242428759223961453738