| L(s) = 1 | − 2i·3-s + (1 − i)5-s + (−1 + i)7-s − 9-s + (3 − 3i)11-s + (3 + 2i)13-s + (−2 − 2i)15-s − 4i·17-s + (−3 − 3i)19-s + (2 + 2i)21-s + 3i·25-s − 4i·27-s + 6·29-s + (−3 − 3i)31-s + (−6 − 6i)33-s + ⋯ |
| L(s) = 1 | − 1.15i·3-s + (0.447 − 0.447i)5-s + (−0.377 + 0.377i)7-s − 0.333·9-s + (0.904 − 0.904i)11-s + (0.832 + 0.554i)13-s + (−0.516 − 0.516i)15-s − 0.970i·17-s + (−0.688 − 0.688i)19-s + (0.436 + 0.436i)21-s + 0.600i·25-s − 0.769i·27-s + 1.11·29-s + (−0.538 − 0.538i)31-s + (−1.04 − 1.04i)33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.289 + 0.957i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.289 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.00220 - 1.35058i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.00220 - 1.35058i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 + (-3 - 2i)T \) |
| good | 3 | \( 1 + 2iT - 3T^{2} \) |
| 5 | \( 1 + (-1 + i)T - 5iT^{2} \) |
| 7 | \( 1 + (1 - i)T - 7iT^{2} \) |
| 11 | \( 1 + (-3 + 3i)T - 11iT^{2} \) |
| 17 | \( 1 + 4iT - 17T^{2} \) |
| 19 | \( 1 + (3 + 3i)T + 19iT^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + (3 + 3i)T + 31iT^{2} \) |
| 37 | \( 1 + (3 + 3i)T + 37iT^{2} \) |
| 41 | \( 1 + (-1 + i)T - 41iT^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 + (5 - 5i)T - 47iT^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 + (-7 + 7i)T - 59iT^{2} \) |
| 61 | \( 1 + 14T + 61T^{2} \) |
| 67 | \( 1 + (-5 - 5i)T + 67iT^{2} \) |
| 71 | \( 1 + (-5 - 5i)T + 71iT^{2} \) |
| 73 | \( 1 + (-9 - 9i)T + 73iT^{2} \) |
| 79 | \( 1 + 6iT - 79T^{2} \) |
| 83 | \( 1 + (7 + 7i)T + 83iT^{2} \) |
| 89 | \( 1 + (-5 - 5i)T + 89iT^{2} \) |
| 97 | \( 1 + (-13 + 13i)T - 97iT^{2} \) |
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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
−9.708403092534637786804036800911, −8.995431566054248029515318066996, −8.396730058539729071390433655237, −7.21698868902712111422462086917, −6.46051528107077955140448040257, −5.90751983764270213542656122899, −4.62534235882220824563664931861, −3.28913799709141643361790041770, −1.98793623399283218921870429115, −0.895926574942647178341408004396,
1.71074380767192155364519340326, 3.36549589406317037280628271231, 4.01739117261477378838166576793, 4.96637276383498824820393105142, 6.23806116816370123029569142599, 6.73358889978087964893655867521, 8.100381838567916013992286606119, 8.958583613492962289766076424109, 9.903280872059627338439098261168, 10.33194415367299669034685782811
