L(s) = 1 | + (1.73 + 2i)7-s + 2·13-s + 8i·19-s − 5·25-s − 10.3·31-s + 6.92i·37-s + 10.3·43-s + (−1.00 + 6.92i)49-s − 14·61-s + 3.46·67-s − 13.8i·73-s + 4i·79-s + (3.46 + 4i)91-s − 13.8i·97-s + 3.46·103-s + ⋯ |
L(s) = 1 | + (0.654 + 0.755i)7-s + 0.554·13-s + 1.83i·19-s − 25-s − 1.86·31-s + 1.13i·37-s + 1.58·43-s + (−0.142 + 0.989i)49-s − 1.79·61-s + 0.423·67-s − 1.62i·73-s + 0.450i·79-s + (0.363 + 0.419i)91-s − 1.40i·97-s + 0.341·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.436 - 0.899i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.436 - 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.467447006\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.467447006\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-1.73 - 2i)T \) |
good | 5 | \( 1 + 5T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 - 8iT - 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + 10.3T + 31T^{2} \) |
| 37 | \( 1 - 6.92iT - 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 - 10.3T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 - 59T^{2} \) |
| 61 | \( 1 + 14T + 61T^{2} \) |
| 67 | \( 1 - 3.46T + 67T^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + 13.8iT - 73T^{2} \) |
| 79 | \( 1 - 4iT - 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 + 13.8iT - 97T^{2} \) |
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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
−8.677581504196959099947495339732, −7.87763342728104376916579699839, −7.49194387697765393318712401387, −6.16895249003112138026847430187, −5.85786076859312040007306860622, −5.01932708388041304516580445284, −4.05489222235286947751049286689, −3.33328704947412590409297021598, −2.12582251292371223293178145075, −1.42327199973411291946639229461,
0.41307889716314904907502465453, 1.58708634058047786718257389312, 2.58968125582364013823318101683, 3.74379319773414750979862277512, 4.31632737346917584689058785434, 5.21772892173754901226165429290, 5.92037058083739490436620535020, 6.95128665535152032641806836758, 7.41349557237542373473196166129, 8.121027130941576072789545312537