L(s) = 1 | − 0.377i·2-s + (−1.03 − 1.38i)3-s + 1.85·4-s − 2.37·5-s + (−0.523 + 0.392i)6-s + i·7-s − 1.45i·8-s + (−0.840 + 2.87i)9-s + 0.895i·10-s − 6.18·11-s + (−1.93 − 2.57i)12-s − 3.83·13-s + 0.377·14-s + (2.46 + 3.28i)15-s + 3.16·16-s − 0.877·17-s + ⋯ |
L(s) = 1 | − 0.267i·2-s + (−0.599 − 0.800i)3-s + 0.928·4-s − 1.06·5-s + (−0.213 + 0.160i)6-s + 0.377i·7-s − 0.515i·8-s + (−0.280 + 0.959i)9-s + 0.283i·10-s − 1.86·11-s + (−0.557 − 0.742i)12-s − 1.06·13-s + 0.100·14-s + (0.636 + 0.848i)15-s + 0.791·16-s − 0.212·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.776 - 0.630i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.776 - 0.630i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0362082 + 0.102050i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0362082 + 0.102050i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.03 + 1.38i)T \) |
| 7 | \( 1 - iT \) |
| 23 | \( 1 + (-1.16 - 4.65i)T \) |
good | 2 | \( 1 + 0.377iT - 2T^{2} \) |
| 5 | \( 1 + 2.37T + 5T^{2} \) |
| 11 | \( 1 + 6.18T + 11T^{2} \) |
| 13 | \( 1 + 3.83T + 13T^{2} \) |
| 17 | \( 1 + 0.877T + 17T^{2} \) |
| 19 | \( 1 + 3.29iT - 19T^{2} \) |
| 29 | \( 1 + 5.12iT - 29T^{2} \) |
| 31 | \( 1 + 8.93T + 31T^{2} \) |
| 37 | \( 1 - 10.9iT - 37T^{2} \) |
| 41 | \( 1 - 1.00iT - 41T^{2} \) |
| 43 | \( 1 + 2.83iT - 43T^{2} \) |
| 47 | \( 1 - 1.64iT - 47T^{2} \) |
| 53 | \( 1 - 4.73T + 53T^{2} \) |
| 59 | \( 1 + 10.7iT - 59T^{2} \) |
| 61 | \( 1 + 6.29iT - 61T^{2} \) |
| 67 | \( 1 + 2.23iT - 67T^{2} \) |
| 71 | \( 1 - 8.32iT - 71T^{2} \) |
| 73 | \( 1 + 9.99T + 73T^{2} \) |
| 79 | \( 1 + 13.6iT - 79T^{2} \) |
| 83 | \( 1 + 7.43T + 83T^{2} \) |
| 89 | \( 1 - 3.93T + 89T^{2} \) |
| 97 | \( 1 - 11.1iT - 97T^{2} \) |
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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
−10.80569292173040833722408819422, −9.840122059653104506020375044563, −8.194227655035357011326119992638, −7.56598518909306449269903231622, −7.01547022114693793664560514124, −5.72829922625849442142756865495, −4.86063820813649314425430803642, −3.06974268979278509674492524476, −2.12217980033567789635405140981, −0.06257632217405505381095132782,
2.61245489997409099247710698630, 3.82907246455670198267303748654, 5.00879606944841061537805304480, 5.78030073875237557369941515507, 7.21273004673326140501404269978, 7.58597004967783881205946529621, 8.732430675635519719540131645635, 10.18722396382266402788899352185, 10.64831360622729254167935219046, 11.31113448644069623075413364485