| L(s) = 1 | − 6.47i·5-s + 2.36·7-s + 3.31i·11-s − 11.3·13-s + 24.1i·17-s − 27.4·19-s + 28.7i·23-s − 16.8·25-s − 2.39i·29-s − 46.9·31-s − 15.3i·35-s + 69.0·37-s − 13.5i·41-s + 55.5·43-s + 39.7i·47-s + ⋯ |
| L(s) = 1 | − 1.29i·5-s + 0.338·7-s + 0.301i·11-s − 0.869·13-s + 1.41i·17-s − 1.44·19-s + 1.25i·23-s − 0.675·25-s − 0.0824i·29-s − 1.51·31-s − 0.438i·35-s + 1.86·37-s − 0.330i·41-s + 1.29·43-s + 0.845i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1188 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1188 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.9139729452\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9139729452\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 - 3.31iT \) |
| good | 5 | \( 1 + 6.47iT - 25T^{2} \) |
| 7 | \( 1 - 2.36T + 49T^{2} \) |
| 13 | \( 1 + 11.3T + 169T^{2} \) |
| 17 | \( 1 - 24.1iT - 289T^{2} \) |
| 19 | \( 1 + 27.4T + 361T^{2} \) |
| 23 | \( 1 - 28.7iT - 529T^{2} \) |
| 29 | \( 1 + 2.39iT - 841T^{2} \) |
| 31 | \( 1 + 46.9T + 961T^{2} \) |
| 37 | \( 1 - 69.0T + 1.36e3T^{2} \) |
| 41 | \( 1 + 13.5iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 55.5T + 1.84e3T^{2} \) |
| 47 | \( 1 - 39.7iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 99.2iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 43.6iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 8.08T + 3.72e3T^{2} \) |
| 67 | \( 1 + 7.40T + 4.48e3T^{2} \) |
| 71 | \( 1 + 60.2iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 104.T + 5.32e3T^{2} \) |
| 79 | \( 1 + 56.7T + 6.24e3T^{2} \) |
| 83 | \( 1 - 153. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 40.0iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 54.2T + 9.40e3T^{2} \) |
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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
−9.503153688947069042164663494354, −9.075625699074022242499895878403, −8.083335583579956653411841480880, −7.59139538718321521130313769790, −6.28749538075193273112595749552, −5.47203526335912534238443117833, −4.57048699939380283020332929160, −3.92469935512606735379848981892, −2.25874979047403704281617983921, −1.28257555065241748655262291564,
0.27083942480870941118703987908, 2.23604858236567334205367777504, 2.86323493110281652139644622730, 4.10815694415350421454620042877, 5.05508548053532292459873085521, 6.18209700892717403247261378228, 6.91671004820288118099395522678, 7.55801078073584818945127350333, 8.514018371980498437161808934247, 9.474543444199348279537072228097
