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
−8.036711526515472770353123756877, −7.67771415067219825270942657658, −6.54836680736339996069987499611, −5.97440971680816426545138835439, −5.17285351006806805083530414906, −4.40064059063919626550316417830, −3.17360639237138299437903135496, −2.82254397250739919130189069978, −1.30288202343338596925516312867, −0.12056875026187638384639040697,
1.71838982383836862301348600696, 2.29089405216119942479353105434, 3.55662933307974575407777360332, 4.52014291113320582051759004203, 4.89273061690963016493473311421, 6.06569832121711268710733104812, 6.75896502268162400461601288683, 7.46575109690944520623640904178, 8.039974462378174613382502327094, 9.057696243891015282571035193238