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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.573 + 0.819i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.573 + 0.819i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.139657802\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.139657802\) |
\(L(\frac{3}{2})\) |
|
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
−8.522738888599355687500959694468, −7.942043811709201450107372207656, −7.18346986978341181172083924998, −6.63469360020656869773801203418, −4.90747354269295239494761972660, −4.66175372796005353775736009556, −4.18017057268072479367636070440, −2.87619829618007826494505335609, −1.47787668837549908216221731891, −0.42723872163521850457252646256,
1.28626626205611577494245615879, 2.78521930363225218992522311452, 3.32635980458357065461916178577, 4.44060559140924325425694657309, 5.11449689536196241471962050825, 6.07074093603467410894074247584, 7.06513218115081176353075793544, 7.85114405089066310686165907722, 8.087182656209204372860871859023, 8.892852025079208688820672968649