L(s) = 1 | + (2.82 + 2.82i)2-s + 16.0i·4-s + (52 − 52i)7-s + (−45.2 + 45.2i)8-s − 124. i·11-s + (−183 − 183i)13-s + 294.·14-s − 256.·16-s + (−350. − 350. i)17-s + 2.10e3i·19-s + (352 − 352i)22-s + (−1.81e3 + 1.81e3i)23-s − 1.03e3i·26-s + (832. + 832. i)28-s + 6.99e3·29-s + ⋯ |
L(s) = 1 | + (0.499 + 0.499i)2-s + 0.500i·4-s + (0.401 − 0.401i)7-s + (−0.250 + 0.250i)8-s − 0.310i·11-s + (−0.300 − 0.300i)13-s + 0.401·14-s − 0.250·16-s + (−0.294 − 0.294i)17-s + 1.33i·19-s + (0.155 − 0.155i)22-s + (−0.714 + 0.714i)23-s − 0.300i·26-s + (0.200 + 0.200i)28-s + 1.54·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.374 - 0.927i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.374 - 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(2.334905069\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.334905069\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-2.82 - 2.82i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (-52 + 52i)T - 1.68e4iT^{2} \) |
| 11 | \( 1 + 124. iT - 1.61e5T^{2} \) |
| 13 | \( 1 + (183 + 183i)T + 3.71e5iT^{2} \) |
| 17 | \( 1 + (350. + 350. i)T + 1.41e6iT^{2} \) |
| 19 | \( 1 - 2.10e3iT - 2.47e6T^{2} \) |
| 23 | \( 1 + (1.81e3 - 1.81e3i)T - 6.43e6iT^{2} \) |
| 29 | \( 1 - 6.99e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 6.78e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + (3.72e3 - 3.72e3i)T - 6.93e7iT^{2} \) |
| 41 | \( 1 - 1.22e4iT - 1.15e8T^{2} \) |
| 43 | \( 1 + (3.18e3 + 3.18e3i)T + 1.47e8iT^{2} \) |
| 47 | \( 1 + (-7.76e3 - 7.76e3i)T + 2.29e8iT^{2} \) |
| 53 | \( 1 + (-2.28e4 + 2.28e4i)T - 4.18e8iT^{2} \) |
| 59 | \( 1 + 3.58e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.37e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + (3.40e4 - 3.40e4i)T - 1.35e9iT^{2} \) |
| 71 | \( 1 - 4.80e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 + (-2.98e4 - 2.98e4i)T + 2.07e9iT^{2} \) |
| 79 | \( 1 - 5.93e4iT - 3.07e9T^{2} \) |
| 83 | \( 1 + (-3.81e4 + 3.81e4i)T - 3.93e9iT^{2} \) |
| 89 | \( 1 + 1.70e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + (9.60e4 - 9.60e4i)T - 8.58e9iT^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.52180693712923897713075082023, −9.753280680754488838339892845319, −8.388925650301261308207054974352, −7.87841427831711052170750743566, −6.78046056911968728777764182727, −5.87198585669582634560332588210, −4.84865497172954611395228692688, −3.90514324453486619891727921943, −2.71508134970827262013081669291, −1.16778263234508688776017036118,
0.48979524782549464151845284154, 1.95634958149178792469295721614, 2.83279513606061245539069270527, 4.29773005788543456470195232346, 4.95237443059099131570385810800, 6.16522217850273999102494474260, 7.06567964675805913195739350332, 8.358106568698121681076734423305, 9.147398598187645585782443696141, 10.22956715341008914072327410377