| L(s) = 1 | + 5.65i·11-s + (−3 + 3i)13-s + 4i·19-s + (−2.82 − 2.82i)23-s − 1.41·29-s + 8·31-s + (−7 − 7i)37-s + 1.41i·41-s + (4 − 4i)43-s + (−2.82 + 2.82i)47-s − 7i·49-s + (−8.48 − 8.48i)53-s − 11.3·59-s − 12·61-s + (−8 − 8i)67-s + ⋯ |
| L(s) = 1 | + 1.70i·11-s + (−0.832 + 0.832i)13-s + 0.917i·19-s + (−0.589 − 0.589i)23-s − 0.262·29-s + 1.43·31-s + (−1.15 − 1.15i)37-s + 0.220i·41-s + (0.609 − 0.609i)43-s + (−0.412 + 0.412i)47-s − i·49-s + (−1.16 − 1.16i)53-s − 1.47·59-s − 1.53·61-s + (−0.977 − 0.977i)67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0618i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 + 0.0618i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4157909573\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4157909573\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 7iT^{2} \) |
| 11 | \( 1 - 5.65iT - 11T^{2} \) |
| 13 | \( 1 + (3 - 3i)T - 13iT^{2} \) |
| 17 | \( 1 - 17iT^{2} \) |
| 19 | \( 1 - 4iT - 19T^{2} \) |
| 23 | \( 1 + (2.82 + 2.82i)T + 23iT^{2} \) |
| 29 | \( 1 + 1.41T + 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 + (7 + 7i)T + 37iT^{2} \) |
| 41 | \( 1 - 1.41iT - 41T^{2} \) |
| 43 | \( 1 + (-4 + 4i)T - 43iT^{2} \) |
| 47 | \( 1 + (2.82 - 2.82i)T - 47iT^{2} \) |
| 53 | \( 1 + (8.48 + 8.48i)T + 53iT^{2} \) |
| 59 | \( 1 + 11.3T + 59T^{2} \) |
| 61 | \( 1 + 12T + 61T^{2} \) |
| 67 | \( 1 + (8 + 8i)T + 67iT^{2} \) |
| 71 | \( 1 - 5.65iT - 71T^{2} \) |
| 73 | \( 1 + (-3 + 3i)T - 73iT^{2} \) |
| 79 | \( 1 + 8iT - 79T^{2} \) |
| 83 | \( 1 + (-11.3 - 11.3i)T + 83iT^{2} \) |
| 89 | \( 1 - 7.07T + 89T^{2} \) |
| 97 | \( 1 + (5 + 5i)T + 97iT^{2} \) |
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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
−9.057696243891015282571035193238, −8.039974462378174613382502327094, −7.46575109690944520623640904178, −6.75896502268162400461601288683, −6.06569832121711268710733104812, −4.89273061690963016493473311421, −4.52014291113320582051759004203, −3.55662933307974575407777360332, −2.29089405216119942479353105434, −1.71838982383836862301348600696,
0.12056875026187638384639040697, 1.30288202343338596925516312867, 2.82254397250739919130189069978, 3.17360639237138299437903135496, 4.40064059063919626550316417830, 5.17285351006806805083530414906, 5.97440971680816426545138835439, 6.54836680736339996069987499611, 7.67771415067219825270942657658, 8.036711526515472770353123756877
