L(s) = 1 | − 0.289i·2-s + i·3-s + 1.91·4-s + 0.289·6-s + 4.91i·7-s − 1.13i·8-s − 9-s + 4.91·11-s + 1.91i·12-s − i·13-s + 1.42·14-s + 3.50·16-s − 4.33i·17-s + 0.289i·18-s − 2.57·19-s + ⋯ |
L(s) = 1 | − 0.204i·2-s + 0.577i·3-s + 0.958·4-s + 0.118·6-s + 1.85i·7-s − 0.400i·8-s − 0.333·9-s + 1.48·11-s + 0.553i·12-s − 0.277i·13-s + 0.379·14-s + 0.876·16-s − 1.05i·17-s + 0.0681i·18-s − 0.591·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.77894 + 1.09944i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.77894 + 1.09944i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 + iT \) |
good | 2 | \( 1 + 0.289iT - 2T^{2} \) |
| 7 | \( 1 - 4.91iT - 7T^{2} \) |
| 11 | \( 1 - 4.91T + 11T^{2} \) |
| 17 | \( 1 + 4.33iT - 17T^{2} \) |
| 19 | \( 1 + 2.57T + 19T^{2} \) |
| 23 | \( 1 - 6.33iT - 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 - 1.42T + 31T^{2} \) |
| 37 | \( 1 - 9.49iT - 37T^{2} \) |
| 41 | \( 1 - 4.33T + 41T^{2} \) |
| 43 | \( 1 - 1.15iT - 43T^{2} \) |
| 47 | \( 1 + 5.42iT - 47T^{2} \) |
| 53 | \( 1 - 0.338iT - 53T^{2} \) |
| 59 | \( 1 - 11.2T + 59T^{2} \) |
| 61 | \( 1 + 10.1T + 61T^{2} \) |
| 67 | \( 1 + 7.25iT - 67T^{2} \) |
| 71 | \( 1 - 0.916T + 71T^{2} \) |
| 73 | \( 1 - 3.15iT - 73T^{2} \) |
| 79 | \( 1 - 3.49T + 79T^{2} \) |
| 83 | \( 1 + 11.2iT - 83T^{2} \) |
| 89 | \( 1 - 0.338T + 89T^{2} \) |
| 97 | \( 1 + 12.3iT - 97T^{2} \) |
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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
−10.02366877665213065412515847565, −9.332503435059081301630500741281, −8.770141544113495597176885643116, −7.66286784669392185369210752828, −6.54701416554594182821886254658, −5.89716524097191964511446655192, −5.04415825064435219545266126214, −3.64011536411592464163751034647, −2.76368165776181243297415920056, −1.71510952892384484279799066133,
1.02974877031961659798242959458, 2.06496727807127902582103824105, 3.66222883397992151075454980071, 4.29599704310334540860921140291, 5.98708061136611726569487958809, 6.61803425444271003609486542931, 7.18038780891415662616736044899, 7.912368761884266965118953834710, 8.888696811246265815702843818650, 10.04824659870449232576108486949