L(s) = 1 | + (0.382 + 0.923i)3-s + (1.60 + 0.666i)5-s + (0.589 + 0.589i)7-s + (−0.707 + 0.707i)9-s + (0.657 − 1.58i)11-s + (3.87 − 1.60i)13-s + 1.74i·15-s + 7.96i·17-s + (−3.97 + 1.64i)19-s + (−0.319 + 0.770i)21-s + (0.452 − 0.452i)23-s + (−1.39 − 1.39i)25-s + (−0.923 − 0.382i)27-s + (−1.69 − 4.08i)29-s + 9.32·31-s + ⋯ |
L(s) = 1 | + (0.220 + 0.533i)3-s + (0.719 + 0.298i)5-s + (0.222 + 0.222i)7-s + (−0.235 + 0.235i)9-s + (0.198 − 0.478i)11-s + (1.07 − 0.444i)13-s + 0.449i·15-s + 1.93i·17-s + (−0.911 + 0.377i)19-s + (−0.0696 + 0.168i)21-s + (0.0942 − 0.0942i)23-s + (−0.278 − 0.278i)25-s + (−0.177 − 0.0736i)27-s + (−0.314 − 0.758i)29-s + 1.67·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.673 - 0.739i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.673 - 0.739i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.53604 + 0.678915i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.53604 + 0.678915i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.382 - 0.923i)T \) |
good | 5 | \( 1 + (-1.60 - 0.666i)T + (3.53 + 3.53i)T^{2} \) |
| 7 | \( 1 + (-0.589 - 0.589i)T + 7iT^{2} \) |
| 11 | \( 1 + (-0.657 + 1.58i)T + (-7.77 - 7.77i)T^{2} \) |
| 13 | \( 1 + (-3.87 + 1.60i)T + (9.19 - 9.19i)T^{2} \) |
| 17 | \( 1 - 7.96iT - 17T^{2} \) |
| 19 | \( 1 + (3.97 - 1.64i)T + (13.4 - 13.4i)T^{2} \) |
| 23 | \( 1 + (-0.452 + 0.452i)T - 23iT^{2} \) |
| 29 | \( 1 + (1.69 + 4.08i)T + (-20.5 + 20.5i)T^{2} \) |
| 31 | \( 1 - 9.32T + 31T^{2} \) |
| 37 | \( 1 + (-0.810 - 0.335i)T + (26.1 + 26.1i)T^{2} \) |
| 41 | \( 1 + (6.65 - 6.65i)T - 41iT^{2} \) |
| 43 | \( 1 + (-2.22 + 5.36i)T + (-30.4 - 30.4i)T^{2} \) |
| 47 | \( 1 - 8.50iT - 47T^{2} \) |
| 53 | \( 1 + (-1.10 + 2.67i)T + (-37.4 - 37.4i)T^{2} \) |
| 59 | \( 1 + (-1.92 - 0.796i)T + (41.7 + 41.7i)T^{2} \) |
| 61 | \( 1 + (4.70 + 11.3i)T + (-43.1 + 43.1i)T^{2} \) |
| 67 | \( 1 + (4.54 + 10.9i)T + (-47.3 + 47.3i)T^{2} \) |
| 71 | \( 1 + (9.09 + 9.09i)T + 71iT^{2} \) |
| 73 | \( 1 + (-1.65 + 1.65i)T - 73iT^{2} \) |
| 79 | \( 1 - 0.580iT - 79T^{2} \) |
| 83 | \( 1 + (3.33 - 1.38i)T + (58.6 - 58.6i)T^{2} \) |
| 89 | \( 1 + (4.91 + 4.91i)T + 89iT^{2} \) |
| 97 | \( 1 + 3.30T + 97T^{2} \) |
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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
−11.17919080825894011603632073545, −10.49546107392731658385664651259, −9.804804343204230169944398935866, −8.512133993928151919073798966071, −8.186082236943887328429826294862, −6.25263229771204614960884882556, −5.97769358621226615027602693354, −4.41612634120950584676875851232, −3.33583502579182624908182522971, −1.85334567456414433462445430272,
1.32836254016752559700746225881, 2.67207987768558633396039350133, 4.28141003545465046591907135055, 5.43999792318486873695551877261, 6.58208600332609589605562638579, 7.33446481367944306301499000905, 8.608734518570373775052389751183, 9.235254717228249573627289476909, 10.23389636471032257201291243028, 11.36341134453651663560926929909