| L(s) = 1 | − 2i·7-s + 2i·11-s + (3 − 2i)13-s − 2·17-s − 6i·19-s + 5·25-s + 2·29-s − 2i·31-s − 4i·37-s − 4i·41-s − 4·43-s − 10i·47-s + 3·49-s + 10·53-s − 6i·59-s + ⋯ |
| L(s) = 1 | − 0.755i·7-s + 0.603i·11-s + (0.832 − 0.554i)13-s − 0.485·17-s − 1.37i·19-s + 25-s + 0.371·29-s − 0.359i·31-s − 0.657i·37-s − 0.624i·41-s − 0.609·43-s − 1.45i·47-s + 0.428·49-s + 1.37·53-s − 0.781i·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.554 + 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.554 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.34243 - 0.718450i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.34243 - 0.718450i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 + (-3 + 2i)T \) |
| good | 5 | \( 1 - 5T^{2} \) |
| 7 | \( 1 + 2iT - 7T^{2} \) |
| 11 | \( 1 - 2iT - 11T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 + 6iT - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 + 2iT - 31T^{2} \) |
| 37 | \( 1 + 4iT - 37T^{2} \) |
| 41 | \( 1 + 4iT - 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 + 10iT - 47T^{2} \) |
| 53 | \( 1 - 10T + 53T^{2} \) |
| 59 | \( 1 + 6iT - 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 - 2iT - 67T^{2} \) |
| 71 | \( 1 - 6iT - 71T^{2} \) |
| 73 | \( 1 - 16iT - 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + 14iT - 83T^{2} \) |
| 89 | \( 1 - 12iT - 89T^{2} \) |
| 97 | \( 1 + 8iT - 97T^{2} \) |
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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
−10.05556937730854702932634338023, −9.016938627770466958580912140167, −8.354400690537480547910025185359, −7.18992766316730570809840454814, −6.76164621931765048789004850993, −5.50987746641886113996009952315, −4.56226852341705740807981827296, −3.63526271034145369197284311853, −2.37998090844589418038291016335, −0.792661949227927141809243847829,
1.42271896864244529902239216190, 2.78847006904093885238101436909, 3.83989480413046097008003456026, 4.96831631312674830744749608159, 6.00577366185051877789225735001, 6.55815987717255360103573849345, 7.82960521544611719930566993790, 8.632519951773996322968238994211, 9.154172297030635505808345985472, 10.26745867203077133794243070967
