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 | \( 1 + (9.60e4 + 9.60e4i)T + 8.58e9iT^{2} \) |
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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
−10.22956715341008914072327410377, −9.147398598187645585782443696141, −8.358106568698121681076734423305, −7.06567964675805913195739350332, −6.16522217850273999102494474260, −4.95237443059099131570385810800, −4.29773005788543456470195232346, −2.83279513606061245539069270527, −1.95634958149178792469295721614, −0.48979524782549464151845284154,
1.16778263234508688776017036118, 2.71508134970827262013081669291, 3.90514324453486619891727921943, 4.84865497172954611395228692688, 5.87198585669582634560332588210, 6.78046056911968728777764182727, 7.87841427831711052170750743566, 8.388925650301261308207054974352, 9.753280680754488838339892845319, 10.52180693712923897713075082023