L(s) = 1 | + (1.42 + 0.979i)3-s − 1.68·5-s + (−0.0236 + 2.64i)7-s + (1.07 + 2.79i)9-s + 3.90i·11-s + (5.24 − 3.02i)13-s + (−2.40 − 1.65i)15-s + (0.201 + 0.348i)17-s + (−0.145 − 0.0840i)19-s + (−2.62 + 3.75i)21-s − 8.88i·23-s − 2.15·25-s + (−1.20 + 5.05i)27-s + (−6.15 − 3.55i)29-s + (5.44 + 3.14i)31-s + ⋯ |
L(s) = 1 | + (0.824 + 0.565i)3-s − 0.753·5-s + (−0.00893 + 0.999i)7-s + (0.359 + 0.932i)9-s + 1.17i·11-s + (1.45 − 0.839i)13-s + (−0.621 − 0.426i)15-s + (0.0488 + 0.0845i)17-s + (−0.0334 − 0.0192i)19-s + (−0.573 + 0.819i)21-s − 1.85i·23-s − 0.431·25-s + (−0.230 + 0.972i)27-s + (−1.14 − 0.659i)29-s + (0.977 + 0.564i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.448 - 0.893i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.448 - 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.24630 + 0.768569i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.24630 + 0.768569i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.42 - 0.979i)T \) |
| 7 | \( 1 + (0.0236 - 2.64i)T \) |
good | 5 | \( 1 + 1.68T + 5T^{2} \) |
| 11 | \( 1 - 3.90iT - 11T^{2} \) |
| 13 | \( 1 + (-5.24 + 3.02i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-0.201 - 0.348i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.145 + 0.0840i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + 8.88iT - 23T^{2} \) |
| 29 | \( 1 + (6.15 + 3.55i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-5.44 - 3.14i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.13 + 5.42i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (1.64 + 2.85i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-1.80 + 3.12i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-4.38 - 7.59i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (4.94 - 2.85i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-2.25 + 3.89i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.43 + 2.56i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.95 + 5.11i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 11.4iT - 71T^{2} \) |
| 73 | \( 1 + (6.05 - 3.49i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (0.603 + 1.04i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (0.181 - 0.314i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-1.38 + 2.39i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-0.508 - 0.293i)T + (48.5 + 84.0i)T^{2} \) |
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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
−12.33907185072460815160125269098, −11.15185121995553693654925426519, −10.26921850899306485311757027849, −9.176646690521282860638692444677, −8.381583597777213422693511857087, −7.62738859293612209681826347604, −6.09805745285482157698846509905, −4.70956253744981472867753765562, −3.66484422654771311288356460803, −2.33043968662820976675585624333,
1.26248103100685459051445891718, 3.40869552281640422237334656336, 3.99840559474892203264954145202, 6.01555412792197454101702752297, 7.11233352737623891879106522318, 7.971055614116770042641918375815, 8.737039800649670170257891401432, 9.829409976418735231889611324274, 11.28134968485632636937714631937, 11.60292054978224055423998911051