| L(s) = 1 | − 1.41i·2-s − 2.00·4-s − 3.19i·5-s + 2.64·7-s + 2.82i·8-s − 4.52·10-s − 18.8i·11-s + 10.3·13-s − 3.74i·14-s + 4.00·16-s + 24.4i·17-s − 26.5·19-s + 6.39i·20-s − 26.6·22-s − 40.8i·23-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.500·4-s − 0.639i·5-s + 0.377·7-s + 0.353i·8-s − 0.452·10-s − 1.71i·11-s + 0.797·13-s − 0.267i·14-s + 0.250·16-s + 1.43i·17-s − 1.39·19-s + 0.319i·20-s − 1.21·22-s − 1.77i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.347313206\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.347313206\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 1.41iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 - 2.64T \) |
| good | 5 | \( 1 + 3.19iT - 25T^{2} \) |
| 11 | \( 1 + 18.8iT - 121T^{2} \) |
| 13 | \( 1 - 10.3T + 169T^{2} \) |
| 17 | \( 1 - 24.4iT - 289T^{2} \) |
| 19 | \( 1 + 26.5T + 361T^{2} \) |
| 23 | \( 1 + 40.8iT - 529T^{2} \) |
| 29 | \( 1 + 21.7iT - 841T^{2} \) |
| 31 | \( 1 + 2.75T + 961T^{2} \) |
| 37 | \( 1 - 45.6T + 1.36e3T^{2} \) |
| 41 | \( 1 - 14.8iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 28.9T + 1.84e3T^{2} \) |
| 47 | \( 1 + 29.7iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 18.1iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 53.5iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 27.8T + 3.72e3T^{2} \) |
| 67 | \( 1 + 92.5T + 4.48e3T^{2} \) |
| 71 | \( 1 + 81.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 81.2T + 5.32e3T^{2} \) |
| 79 | \( 1 + 84.6T + 6.24e3T^{2} \) |
| 83 | \( 1 + 115. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 13.7iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 111.T + 9.40e3T^{2} \) |
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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
−8.934249755400794819368249345846, −8.539301772601925104388320460035, −8.094416143677723267072517603882, −6.34235164949820831664305067564, −5.90101164743038867646638505661, −4.59869648715312073728903230810, −3.94316382431310164021542976201, −2.79069329458208779015205341249, −1.50740272311362489504688238076, −0.42118622533437534081552578665,
1.55383011476896633044079782890, 2.86636020525034751701132586392, 4.15602397543941288553362276515, 4.90592438285963334030217458357, 5.90346561885863703311640120489, 6.99219604029860926174551192092, 7.25971392172944610420573372402, 8.264754130405146206277784017062, 9.236735466658538309859216416065, 9.842463115713834849689420167089
