L(s) = 1 | + (0.292 − 0.707i)3-s + (−2.70 + 1.12i)5-s + (1 − i)7-s + (1.70 + 1.70i)9-s + (1.70 + 4.12i)11-s + (−0.707 − 0.292i)13-s + 2.24i·15-s − 2.82i·17-s + (3.70 + 1.53i)19-s + (−0.414 − i)21-s + (5.82 + 5.82i)23-s + (2.53 − 2.53i)25-s + (3.82 − 1.58i)27-s + (−1.29 + 3.12i)29-s + 4·31-s + ⋯ |
L(s) = 1 | + (0.169 − 0.408i)3-s + (−1.21 + 0.501i)5-s + (0.377 − 0.377i)7-s + (0.569 + 0.569i)9-s + (0.514 + 1.24i)11-s + (−0.196 − 0.0812i)13-s + 0.579i·15-s − 0.685i·17-s + (0.850 + 0.352i)19-s + (−0.0903 − 0.218i)21-s + (1.21 + 1.21i)23-s + (0.507 − 0.507i)25-s + (0.736 − 0.305i)27-s + (−0.240 + 0.579i)29-s + 0.718·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.831 - 0.555i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.831 - 0.555i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.27905 + 0.387996i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.27905 + 0.387996i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
good | 3 | \( 1 + (-0.292 + 0.707i)T + (-2.12 - 2.12i)T^{2} \) |
| 5 | \( 1 + (2.70 - 1.12i)T + (3.53 - 3.53i)T^{2} \) |
| 7 | \( 1 + (-1 + i)T - 7iT^{2} \) |
| 11 | \( 1 + (-1.70 - 4.12i)T + (-7.77 + 7.77i)T^{2} \) |
| 13 | \( 1 + (0.707 + 0.292i)T + (9.19 + 9.19i)T^{2} \) |
| 17 | \( 1 + 2.82iT - 17T^{2} \) |
| 19 | \( 1 + (-3.70 - 1.53i)T + (13.4 + 13.4i)T^{2} \) |
| 23 | \( 1 + (-5.82 - 5.82i)T + 23iT^{2} \) |
| 29 | \( 1 + (1.29 - 3.12i)T + (-20.5 - 20.5i)T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 + (0.707 - 0.292i)T + (26.1 - 26.1i)T^{2} \) |
| 41 | \( 1 + (-0.171 - 0.171i)T + 41iT^{2} \) |
| 43 | \( 1 + (1.94 + 4.70i)T + (-30.4 + 30.4i)T^{2} \) |
| 47 | \( 1 - 0.343iT - 47T^{2} \) |
| 53 | \( 1 + (0.464 + 1.12i)T + (-37.4 + 37.4i)T^{2} \) |
| 59 | \( 1 + (4.53 - 1.87i)T + (41.7 - 41.7i)T^{2} \) |
| 61 | \( 1 + (-0.707 + 1.70i)T + (-43.1 - 43.1i)T^{2} \) |
| 67 | \( 1 + (-2.29 + 5.53i)T + (-47.3 - 47.3i)T^{2} \) |
| 71 | \( 1 + (5.82 - 5.82i)T - 71iT^{2} \) |
| 73 | \( 1 + (7 + 7i)T + 73iT^{2} \) |
| 79 | \( 1 + 6iT - 79T^{2} \) |
| 83 | \( 1 + (-4.53 - 1.87i)T + (58.6 + 58.6i)T^{2} \) |
| 89 | \( 1 + (8.65 - 8.65i)T - 89iT^{2} \) |
| 97 | \( 1 + 18.4T + 97T^{2} \) |
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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
−11.11234603951255219152222071747, −10.15559015367511798280908907916, −9.232489022891668025389669654773, −7.897825510294918740728154580855, −7.35236703304040547598738008067, −6.93561384210194964130756690710, −5.10815283597588213893067251017, −4.25366720111004955444283823362, −3.09249657116822073639937590320, −1.48359960357213447278970742131,
0.921255956001779853244271660467, 3.10067551114759327246420915122, 4.05947621191648753863234279951, 4.88388952488504719873670699192, 6.20719131419109738494614294249, 7.28788736992689290156643864222, 8.420232873331367739472059293273, 8.777315268275063835248548165301, 9.850284588790431991023751830651, 11.00250896097147893343433686511