| L(s) = 1 | − 7.59i·5-s + 2.64·7-s + 15.9i·11-s − 13.6·13-s + 32.2i·17-s + 23.4·19-s + 7.96i·23-s − 32.7·25-s − 23.0i·29-s − 29.9·31-s − 20.1i·35-s − 27.4·37-s − 19.8i·41-s − 19.9·43-s − 17.4i·47-s + ⋯ |
| L(s) = 1 | − 1.51i·5-s + 0.377·7-s + 1.45i·11-s − 1.05·13-s + 1.89i·17-s + 1.23·19-s + 0.346i·23-s − 1.30·25-s − 0.794i·29-s − 0.967·31-s − 0.574i·35-s − 0.743·37-s − 0.483i·41-s − 0.463·43-s − 0.370i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.187707305\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.187707305\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - 2.64T \) |
| good | 5 | \( 1 + 7.59iT - 25T^{2} \) |
| 11 | \( 1 - 15.9iT - 121T^{2} \) |
| 13 | \( 1 + 13.6T + 169T^{2} \) |
| 17 | \( 1 - 32.2iT - 289T^{2} \) |
| 19 | \( 1 - 23.4T + 361T^{2} \) |
| 23 | \( 1 - 7.96iT - 529T^{2} \) |
| 29 | \( 1 + 23.0iT - 841T^{2} \) |
| 31 | \( 1 + 29.9T + 961T^{2} \) |
| 37 | \( 1 + 27.4T + 1.36e3T^{2} \) |
| 41 | \( 1 + 19.8iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 19.9T + 1.84e3T^{2} \) |
| 47 | \( 1 + 17.4iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 38.0iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 108. iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 36.4T + 3.72e3T^{2} \) |
| 67 | \( 1 - 88.2T + 4.48e3T^{2} \) |
| 71 | \( 1 + 93.7iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 41.3T + 5.32e3T^{2} \) |
| 79 | \( 1 - 58.8T + 6.24e3T^{2} \) |
| 83 | \( 1 + 112. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 47.5iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 157.T + 9.40e3T^{2} \) |
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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
−7.988708578125821217893485377981, −7.50970188475131168493016630978, −6.59596067404094087392267447968, −5.44342151856866523308371390545, −5.08133656483321586691115289260, −4.35223345853133690575814025817, −3.58059583938357035783741257057, −1.98036306530490486720982154253, −1.62244216078169191635409964875, −0.26781032095522049955209016194,
0.978978764275975843488107164826, 2.50283985330256399625486609750, 2.95787815670899389948222955901, 3.67447366862711858394352991404, 4.98576944478579904692383817759, 5.49299711441042513740330012089, 6.45292684714696923603336765685, 7.28780682918312157955625636585, 7.41732607925785350022317294042, 8.493998373458090327921489294149
