L(s) = 1 | + 5.83·5-s − 2.82·7-s − 16.4·11-s + 8.24i·13-s + 7.07i·17-s − 23.3i·19-s − 20i·23-s + 9·25-s − 29.1·29-s − 42.4·31-s − 16.4·35-s − 49.4i·37-s − 26.8i·41-s − 44i·47-s − 41·49-s + ⋯ |
L(s) = 1 | + 1.16·5-s − 0.404·7-s − 1.49·11-s + 0.634i·13-s + 0.415i·17-s − 1.22i·19-s − 0.869i·23-s + 0.359·25-s − 1.00·29-s − 1.36·31-s − 0.471·35-s − 1.33i·37-s − 0.655i·41-s − 0.936i·47-s − 0.836·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 + 0.577i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.6213324261\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6213324261\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 - 5.83T + 25T^{2} \) |
| 7 | \( 1 + 2.82T + 49T^{2} \) |
| 11 | \( 1 + 16.4T + 121T^{2} \) |
| 13 | \( 1 - 8.24iT - 169T^{2} \) |
| 17 | \( 1 - 7.07iT - 289T^{2} \) |
| 19 | \( 1 + 23.3iT - 361T^{2} \) |
| 23 | \( 1 + 20iT - 529T^{2} \) |
| 29 | \( 1 + 29.1T + 841T^{2} \) |
| 31 | \( 1 + 42.4T + 961T^{2} \) |
| 37 | \( 1 + 49.4iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 26.8iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 1.84e3T^{2} \) |
| 47 | \( 1 + 44iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 29.1T + 2.80e3T^{2} \) |
| 59 | \( 1 - 65.9T + 3.48e3T^{2} \) |
| 61 | \( 1 - 82.4iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 116. iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 100iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 40T + 5.32e3T^{2} \) |
| 79 | \( 1 + 127.T + 6.24e3T^{2} \) |
| 83 | \( 1 - 82.4T + 6.88e3T^{2} \) |
| 89 | \( 1 + 114. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 40T + 9.40e3T^{2} \) |
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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
−9.256449078085819327022134842520, −8.700553151058440100341032766230, −7.48501839787980798812481181451, −6.78229191576617977562868628979, −5.72906329625926520939413308339, −5.24635713009729732045551616602, −4.00158764862590512806330958637, −2.65229207769409095821805644027, −1.96432654080240081258775579322, −0.16562021406161888191630680934,
1.59085298506657218532344577161, 2.64699676082283425349733777957, 3.60378502237672287487362315844, 5.19234346364699331393581288650, 5.55869626738904357633192806583, 6.45035466885515745544080513191, 7.59287280677415448765256671113, 8.167833931519723952607625776601, 9.444693615441779250766576901478, 9.827901992612153828103571704865