L(s) = 1 | + (−1.15 + 1.29i)3-s − 5-s − 2.00i·7-s + (−0.351 − 2.97i)9-s + 4.96·11-s − 1.09i·13-s + (1.15 − 1.29i)15-s + 2.56i·17-s − 2.48·19-s + (2.59 + 2.31i)21-s + 2.68i·23-s + 25-s + (4.26 + 2.97i)27-s + 0.905i·29-s − 0.150i·31-s + ⋯ |
L(s) = 1 | + (−0.664 + 0.747i)3-s − 0.447·5-s − 0.758i·7-s + (−0.117 − 0.993i)9-s + 1.49·11-s − 0.303i·13-s + (0.297 − 0.334i)15-s + 0.622i·17-s − 0.570·19-s + (0.567 + 0.504i)21-s + 0.560i·23-s + 0.200·25-s + (0.820 + 0.572i)27-s + 0.168i·29-s − 0.0270i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.857 + 0.515i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.857 + 0.515i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.222475681\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.222475681\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.15 - 1.29i)T \) |
| 5 | \( 1 + T \) |
| 67 | \( 1 + (1.50 + 8.04i)T \) |
good | 7 | \( 1 + 2.00iT - 7T^{2} \) |
| 11 | \( 1 - 4.96T + 11T^{2} \) |
| 13 | \( 1 + 1.09iT - 13T^{2} \) |
| 17 | \( 1 - 2.56iT - 17T^{2} \) |
| 19 | \( 1 + 2.48T + 19T^{2} \) |
| 23 | \( 1 - 2.68iT - 23T^{2} \) |
| 29 | \( 1 - 0.905iT - 29T^{2} \) |
| 31 | \( 1 + 0.150iT - 31T^{2} \) |
| 37 | \( 1 + 8.64T + 37T^{2} \) |
| 41 | \( 1 - 9.39T + 41T^{2} \) |
| 43 | \( 1 - 1.68iT - 43T^{2} \) |
| 47 | \( 1 + 13.1iT - 47T^{2} \) |
| 53 | \( 1 - 1.64T + 53T^{2} \) |
| 59 | \( 1 - 7.45iT - 59T^{2} \) |
| 61 | \( 1 + 11.3iT - 61T^{2} \) |
| 71 | \( 1 + 0.805iT - 71T^{2} \) |
| 73 | \( 1 + 15.9T + 73T^{2} \) |
| 79 | \( 1 - 0.585iT - 79T^{2} \) |
| 83 | \( 1 + 8.06iT - 83T^{2} \) |
| 89 | \( 1 - 18.0iT - 89T^{2} \) |
| 97 | \( 1 + 6.66iT - 97T^{2} \) |
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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
−8.591091168200577087979945610672, −7.51520649948079951809315643768, −6.82848717804113905965678321389, −6.19470435067541900585159145100, −5.37774006514614976629170231853, −4.39508932038266259934208890987, −3.91905796926497179926769625798, −3.30594146127388218905294003101, −1.62945072983580175477222657526, −0.51313907554585020323012671864,
0.898504420598864332490836833684, 1.93346512762620488781845595647, 2.87309999965461716545932666030, 4.09950138902247221954477124150, 4.73025214141822030944751567054, 5.78588797712520871127133959007, 6.26621488665940664910378743269, 7.04188522456357960905131349929, 7.56209806778095591401028916334, 8.715216583174692453750450036730