L(s) = 1 | + 10.5i·3-s − 16.8·5-s − 38.6·7-s − 31.1·9-s − 70.8·11-s − 231. i·13-s − 178. i·15-s − 90.4·17-s + (−319. + 167. i)19-s − 409. i·21-s − 285.·23-s − 342.·25-s + 527. i·27-s + 285. i·29-s + 852. i·31-s + ⋯ |
L(s) = 1 | + 1.17i·3-s − 0.672·5-s − 0.788·7-s − 0.384·9-s − 0.585·11-s − 1.36i·13-s − 0.791i·15-s − 0.312·17-s + (−0.885 + 0.464i)19-s − 0.928i·21-s − 0.539·23-s − 0.547·25-s + 0.723i·27-s + 0.340i·29-s + 0.886i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.885 + 0.464i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.885 + 0.464i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(0.0633840 - 0.257492i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0633840 - 0.257492i\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 + (319. - 167. i)T \) |
good | 3 | \( 1 - 10.5iT - 81T^{2} \) |
| 5 | \( 1 + 16.8T + 625T^{2} \) |
| 7 | \( 1 + 38.6T + 2.40e3T^{2} \) |
| 11 | \( 1 + 70.8T + 1.46e4T^{2} \) |
| 13 | \( 1 + 231. iT - 2.85e4T^{2} \) |
| 17 | \( 1 + 90.4T + 8.35e4T^{2} \) |
| 23 | \( 1 + 285.T + 2.79e5T^{2} \) |
| 29 | \( 1 - 285. iT - 7.07e5T^{2} \) |
| 31 | \( 1 - 852. iT - 9.23e5T^{2} \) |
| 37 | \( 1 + 54.8iT - 1.87e6T^{2} \) |
| 41 | \( 1 + 37.4iT - 2.82e6T^{2} \) |
| 43 | \( 1 - 691.T + 3.41e6T^{2} \) |
| 47 | \( 1 - 2.63e3T + 4.87e6T^{2} \) |
| 53 | \( 1 - 3.83e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 - 3.33e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 + 2.87e3T + 1.38e7T^{2} \) |
| 67 | \( 1 - 4.05e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 + 3.31e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 5.54e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + 1.03e4iT - 3.89e7T^{2} \) |
| 83 | \( 1 + 3.54e3T + 4.74e7T^{2} \) |
| 89 | \( 1 + 1.32e4iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 4.28e3iT - 8.85e7T^{2} \) |
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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
−14.74950813513829660176427555464, −13.20032761822566803960060175081, −12.24190169438572911176211412970, −10.68672282609788604624290366783, −10.18001929595896627997168867891, −8.852858990326135310209763223777, −7.57388684701246057325189250157, −5.81122963488803458951286820947, −4.34940345660256481872129593530, −3.17090468470234517188846609694,
0.12864074262186600805070008438, 2.19370753229739653064038748069, 4.15543513150220231475125543116, 6.25529918198481176347630640529, 7.15824741785784744691476254349, 8.254643720862422231952712118985, 9.652674119096762551192884048963, 11.22030079684402263864707303921, 12.20531326986528623615469092223, 13.07222267280321525673363674377