| L(s) = 1 | + (−1.41 − 1.73i)5-s + 2.82i·7-s − 4.89·11-s + 4.89i·13-s − 3.46i·17-s − 6.92·19-s − 4i·23-s + (−0.999 + 4.89i)25-s + 8.48·29-s + 6.92·31-s + (4.89 − 4.00i)35-s − 4.89i·37-s + 5.65·41-s − 11.3i·43-s + 4i·47-s + ⋯ |
| L(s) = 1 | + (−0.632 − 0.774i)5-s + 1.06i·7-s − 1.47·11-s + 1.35i·13-s − 0.840i·17-s − 1.58·19-s − 0.834i·23-s + (−0.199 + 0.979i)25-s + 1.57·29-s + 1.24·31-s + (0.828 − 0.676i)35-s − 0.805i·37-s + 0.883·41-s − 1.72i·43-s + 0.583i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.632 + 0.774i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.632 + 0.774i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.032325469\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.032325469\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.41 + 1.73i)T \) |
| good | 7 | \( 1 - 2.82iT - 7T^{2} \) |
| 11 | \( 1 + 4.89T + 11T^{2} \) |
| 13 | \( 1 - 4.89iT - 13T^{2} \) |
| 17 | \( 1 + 3.46iT - 17T^{2} \) |
| 19 | \( 1 + 6.92T + 19T^{2} \) |
| 23 | \( 1 + 4iT - 23T^{2} \) |
| 29 | \( 1 - 8.48T + 29T^{2} \) |
| 31 | \( 1 - 6.92T + 31T^{2} \) |
| 37 | \( 1 + 4.89iT - 37T^{2} \) |
| 41 | \( 1 - 5.65T + 41T^{2} \) |
| 43 | \( 1 + 11.3iT - 43T^{2} \) |
| 47 | \( 1 - 4iT - 47T^{2} \) |
| 53 | \( 1 + 3.46iT - 53T^{2} \) |
| 59 | \( 1 - 4.89T + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 + 5.65iT - 67T^{2} \) |
| 71 | \( 1 + 9.79T + 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 - 6.92T + 79T^{2} \) |
| 83 | \( 1 - 16iT - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 9.79iT - 97T^{2} \) |
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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
−8.563443179826702468324754164203, −8.212478509658144953741099125921, −7.17667685012691762311699901214, −6.38906625735589179717028482324, −5.45684716560003813790421698212, −4.71898944757228182201972777808, −4.17270209063145256431569483633, −2.71971137146358184744713632771, −2.14940712530319978284094157716, −0.44684024077548355643986902665,
0.818012293844322017240786681107, 2.51239676130449104717964851748, 3.17221280488497547552003560490, 4.14362127644062651234984522343, 4.83867602564677873618358155812, 5.98671774823206369486106099036, 6.61568118548870260159858543637, 7.58823268433084590967614063744, 7.992474133221564626016249166724, 8.510260017793716758309647494649
