| L(s) = 1 | + 6.72i·2-s − 29.1·4-s − 1.48i·5-s − 13.7i·7-s − 88.6i·8-s + 9.96·10-s + 10.2·11-s + 98.4·13-s + 92.7·14-s + 128.·16-s + 286.·17-s − 367. i·19-s + 43.2i·20-s + 68.5i·22-s + 242.·23-s + ⋯ |
| L(s) = 1 | + 1.68i·2-s − 1.82·4-s − 0.0592i·5-s − 0.281i·7-s − 1.38i·8-s + 0.0996·10-s + 0.0843·11-s + 0.582·13-s + 0.473·14-s + 0.503·16-s + 0.991·17-s − 1.01i·19-s + 0.108i·20-s + 0.141i·22-s + 0.457·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.836 - 0.548i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.836 - 0.548i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.821550210\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.821550210\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 43 | \( 1 + (1.54e3 + 1.01e3i)T \) |
| good | 2 | \( 1 - 6.72iT - 16T^{2} \) |
| 5 | \( 1 + 1.48iT - 625T^{2} \) |
| 7 | \( 1 + 13.7iT - 2.40e3T^{2} \) |
| 11 | \( 1 - 10.2T + 1.46e4T^{2} \) |
| 13 | \( 1 - 98.4T + 2.85e4T^{2} \) |
| 17 | \( 1 - 286.T + 8.35e4T^{2} \) |
| 19 | \( 1 + 367. iT - 1.30e5T^{2} \) |
| 23 | \( 1 - 242.T + 2.79e5T^{2} \) |
| 29 | \( 1 - 1.14e3iT - 7.07e5T^{2} \) |
| 31 | \( 1 - 895.T + 9.23e5T^{2} \) |
| 37 | \( 1 - 2.29e3iT - 1.87e6T^{2} \) |
| 41 | \( 1 + 1.69e3T + 2.82e6T^{2} \) |
| 47 | \( 1 + 743.T + 4.87e6T^{2} \) |
| 53 | \( 1 + 99.3T + 7.89e6T^{2} \) |
| 59 | \( 1 + 3.28e3T + 1.21e7T^{2} \) |
| 61 | \( 1 - 3.22e3iT - 1.38e7T^{2} \) |
| 67 | \( 1 - 5.55e3T + 2.01e7T^{2} \) |
| 71 | \( 1 - 2.95e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 3.61e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 - 2.38e3T + 3.89e7T^{2} \) |
| 83 | \( 1 - 6.42e3T + 4.74e7T^{2} \) |
| 89 | \( 1 - 3.29e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 9.32e3T + 8.85e7T^{2} \) |
| 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
−10.94412566940568168355790460116, −9.875087748144982281029058095683, −8.798862272427462318907830551432, −8.242773803333756040968086190781, −7.08467918830017323492622303252, −6.58769103969682160634735627630, −5.38134355907145543144310960653, −4.65899140191940907377385720165, −3.22881335146405453979412200360, −0.985852064485153469616146880462,
0.69217930902843848953149460929, 1.82490054733683233381669070392, 3.03876795644483399811647391654, 3.90164744387938728028638440616, 5.10362070044459467404858853769, 6.32837966109396170936982794234, 7.87258849402143375695153279624, 8.823998799782689850218857598848, 9.737496910309070931506742787961, 10.42936527590268536735719422405
