L(s) = 1 | + 0.288·3-s − 2.23i·5-s + 12.5i·7-s − 8.91·9-s − 11.3i·11-s + 23.5·13-s − 0.645i·15-s − 15.6i·17-s + 22.3i·19-s + 3.61i·21-s + (−10.9 − 20.2i)23-s − 5.00·25-s − 5.17·27-s − 15.0·29-s + 40.2·31-s + ⋯ |
L(s) = 1 | + 0.0962·3-s − 0.447i·5-s + 1.78i·7-s − 0.990·9-s − 1.03i·11-s + 1.81·13-s − 0.0430i·15-s − 0.919i·17-s + 1.17i·19-s + 0.172i·21-s + (−0.477 − 0.878i)23-s − 0.200·25-s − 0.191·27-s − 0.519·29-s + 1.29·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.878 - 0.477i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.878 - 0.477i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.982045536\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.982045536\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + 2.23iT \) |
| 23 | \( 1 + (10.9 + 20.2i)T \) |
good | 3 | \( 1 - 0.288T + 9T^{2} \) |
| 7 | \( 1 - 12.5iT - 49T^{2} \) |
| 11 | \( 1 + 11.3iT - 121T^{2} \) |
| 13 | \( 1 - 23.5T + 169T^{2} \) |
| 17 | \( 1 + 15.6iT - 289T^{2} \) |
| 19 | \( 1 - 22.3iT - 361T^{2} \) |
| 29 | \( 1 + 15.0T + 841T^{2} \) |
| 31 | \( 1 - 40.2T + 961T^{2} \) |
| 37 | \( 1 + 45.3iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 9.36T + 1.68e3T^{2} \) |
| 43 | \( 1 - 22.3iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 53.8T + 2.20e3T^{2} \) |
| 53 | \( 1 - 81.3iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 49.2T + 3.48e3T^{2} \) |
| 61 | \( 1 - 10.9iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 25.0iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 99.6T + 5.04e3T^{2} \) |
| 73 | \( 1 + 57.0T + 5.32e3T^{2} \) |
| 79 | \( 1 - 102. iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 112. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 81.4iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 47.6iT - 9.40e3T^{2} \) |
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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
−8.834202650824560175280084622836, −8.539282113610337705667294534432, −7.980794999264723539564846669128, −6.36268595440804816707084314813, −5.78754256245091876452152743879, −5.48385405116990005256848140646, −4.06233230064196572151405116369, −3.06924032615083853178675236384, −2.28295087614612330351932137189, −0.861436475748130218811324587539,
0.69058085968253730702455987376, 1.85610592825621344985771396770, 3.28366527962521565391997667372, 3.87606195791749777481488896521, 4.75527638117549201904324152340, 6.01081100424366001696719196319, 6.64490739391786904766803649260, 7.42084255874740811975685199051, 8.159498233107617864895654128449, 8.917211491354122923597308857373