| L(s) = 1 | − 1.73i·3-s − 8.36i·5-s − 2.99·9-s − 6·11-s − 17.8i·13-s − 14.4·15-s − 18.7i·17-s − 17.0i·19-s − 13.4·23-s − 44.9·25-s + 5.19i·27-s + 33.9·29-s − 14.7i·31-s + 10.3i·33-s + 5.97·37-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s − 1.67i·5-s − 0.333·9-s − 0.545·11-s − 1.37i·13-s − 0.965·15-s − 1.10i·17-s − 0.895i·19-s − 0.585·23-s − 1.79·25-s + 0.192i·27-s + 1.17·29-s − 0.475i·31-s + 0.314i·33-s + 0.161·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.654 - 0.755i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.654 - 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.302087317\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.302087317\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 1.73iT \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 8.36iT - 25T^{2} \) |
| 11 | \( 1 + 6T + 121T^{2} \) |
| 13 | \( 1 + 17.8iT - 169T^{2} \) |
| 17 | \( 1 + 18.7iT - 289T^{2} \) |
| 19 | \( 1 + 17.0iT - 361T^{2} \) |
| 23 | \( 1 + 13.4T + 529T^{2} \) |
| 29 | \( 1 - 33.9T + 841T^{2} \) |
| 31 | \( 1 + 14.7iT - 961T^{2} \) |
| 37 | \( 1 - 5.97T + 1.36e3T^{2} \) |
| 41 | \( 1 - 35.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 15.4T + 1.84e3T^{2} \) |
| 47 | \( 1 + 33.2iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 34.5T + 2.80e3T^{2} \) |
| 59 | \( 1 + 27.3iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 40.3iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 114.T + 4.48e3T^{2} \) |
| 71 | \( 1 + 18.6T + 5.04e3T^{2} \) |
| 73 | \( 1 + 117. iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 88.3T + 6.24e3T^{2} \) |
| 83 | \( 1 + 75.7iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 20.7iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 30.5iT - 9.40e3T^{2} \) |
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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
−8.085700348084167912313324210775, −7.930247252952033942723744340911, −6.80615562160224152103478861169, −5.80047357716409569236947456336, −5.09110165261112585239955825547, −4.60636002865501278824636055540, −3.22321917332194049799690789813, −2.24849693810977941160945183763, −0.936650219279052917567870804100, −0.36738784913036126912468234938,
1.82013981704763596400555201247, 2.71469195851598807289828331625, 3.64933270463664985134029901241, 4.25790476428004183197531717691, 5.46558260536548656582654618917, 6.37631159098017289167894409972, 6.76624065069496284385287841507, 7.79651243280209416420020133085, 8.439537512531795573180848212216, 9.501454523937149700431298663732
