L(s) = 1 | − 2.12i·2-s − i·3-s − 2.51·4-s − 2.12·6-s − 3.64i·7-s + 1.09i·8-s − 9-s + 11-s + 2.51i·12-s + 1.51i·13-s − 7.73·14-s − 2.70·16-s − 1.15i·17-s + 2.12i·18-s − 2.60·19-s + ⋯ |
L(s) = 1 | − 1.50i·2-s − 0.577i·3-s − 1.25·4-s − 0.867·6-s − 1.37i·7-s + 0.387i·8-s − 0.333·9-s + 0.301·11-s + 0.726i·12-s + 0.420i·13-s − 2.06·14-s − 0.676·16-s − 0.280i·17-s + 0.500i·18-s − 0.598·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.604938 + 0.978810i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.604938 + 0.978810i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + iT \) |
| 5 | \( 1 \) |
| 11 | \( 1 - T \) |
good | 2 | \( 1 + 2.12iT - 2T^{2} \) |
| 7 | \( 1 + 3.64iT - 7T^{2} \) |
| 13 | \( 1 - 1.51iT - 13T^{2} \) |
| 17 | \( 1 + 1.15iT - 17T^{2} \) |
| 19 | \( 1 + 2.60T + 19T^{2} \) |
| 23 | \( 1 + 5.73iT - 23T^{2} \) |
| 29 | \( 1 + 6.24T + 29T^{2} \) |
| 31 | \( 1 - 5.51T + 31T^{2} \) |
| 37 | \( 1 + 0.454iT - 37T^{2} \) |
| 41 | \( 1 - 4.12T + 41T^{2} \) |
| 43 | \( 1 - 11.7iT - 43T^{2} \) |
| 47 | \( 1 - 3.48iT - 47T^{2} \) |
| 53 | \( 1 + 12.5iT - 53T^{2} \) |
| 59 | \( 1 - 7.73T + 59T^{2} \) |
| 61 | \( 1 + 12.0T + 61T^{2} \) |
| 67 | \( 1 - 14.2iT - 67T^{2} \) |
| 71 | \( 1 - 8.51T + 71T^{2} \) |
| 73 | \( 1 + 9.21iT - 73T^{2} \) |
| 79 | \( 1 + 5.09T + 79T^{2} \) |
| 83 | \( 1 + 14.7iT - 83T^{2} \) |
| 89 | \( 1 - 10.4T + 89T^{2} \) |
| 97 | \( 1 - 6.77iT - 97T^{2} \) |
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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
−9.926791958087958072114896537981, −9.095859944584147206381944421647, −8.050538275006145381608630409565, −7.05315812352007219562708876129, −6.32779562786236021567732250748, −4.63145756213600638102151179305, −3.97935231045098896335378031425, −2.85306763793579772075483243932, −1.68729420040418093409655607667, −0.56744977828891505152320765071,
2.32285850768391606188067586596, 3.77309870844691475126191373343, 4.98381156796781296434293067236, 5.67981715122935171903681675289, 6.25420744733698131452314451541, 7.38926996729282910680261481348, 8.243922113188058498243258141922, 8.957452143185897948066623279841, 9.491270156066729688650409678289, 10.68654892140462194465546155143