| L(s) = 1 | + 1.73i·3-s + 2.63·5-s − 12.5i·7-s − 2.99·9-s − 5.79i·11-s − 8.78·13-s + 4.56i·15-s − 30.1·17-s − 17.4i·19-s + 21.7·21-s − 2.48i·23-s − 18.0·25-s − 5.19i·27-s + 26.4·29-s − 38.0i·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + 0.527·5-s − 1.79i·7-s − 0.333·9-s − 0.527i·11-s − 0.675·13-s + 0.304i·15-s − 1.77·17-s − 0.916i·19-s + 1.03·21-s − 0.108i·23-s − 0.722·25-s − 0.192i·27-s + 0.910·29-s − 1.22i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.905787 - 0.905787i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.905787 - 0.905787i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 1.73iT \) |
| good | 5 | \( 1 - 2.63T + 25T^{2} \) |
| 7 | \( 1 + 12.5iT - 49T^{2} \) |
| 11 | \( 1 + 5.79iT - 121T^{2} \) |
| 13 | \( 1 + 8.78T + 169T^{2} \) |
| 17 | \( 1 + 30.1T + 289T^{2} \) |
| 19 | \( 1 + 17.4iT - 361T^{2} \) |
| 23 | \( 1 + 2.48iT - 529T^{2} \) |
| 29 | \( 1 - 26.4T + 841T^{2} \) |
| 31 | \( 1 + 38.0iT - 961T^{2} \) |
| 37 | \( 1 - 47.7T + 1.36e3T^{2} \) |
| 41 | \( 1 - 53.3T + 1.68e3T^{2} \) |
| 43 | \( 1 - 30.7iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 16.2iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 49.8T + 2.80e3T^{2} \) |
| 59 | \( 1 + 107. iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 62.6T + 3.72e3T^{2} \) |
| 67 | \( 1 - 60.9iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 19.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 5.13T + 5.32e3T^{2} \) |
| 79 | \( 1 - 6.83iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 159. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 39.4T + 7.92e3T^{2} \) |
| 97 | \( 1 + 60.5T + 9.40e3T^{2} \) |
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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
−10.93691323488696277398848129599, −9.961868043936629820282315098828, −9.329720796487291995391110933646, −8.095801546109705616134550207046, −7.06215730503282654625625207003, −6.16500319463084359041677485744, −4.68928606866617477818311187246, −4.06231395579744596012854057051, −2.51679090231655051418073435117, −0.53729943089220466036964789454,
1.94349633181754953799497372324, 2.65666432258528588621905580265, 4.62569662611022377245077039767, 5.74954580577366501300163633558, 6.42242564622013309324138668894, 7.62987830169811916487081601454, 8.751320486584851471005062583695, 9.293395901485764195114302983472, 10.39931445351696337449619585616, 11.63961745336823681745346683256
