| L(s) = 1 | + 5.85i·3-s − 5.78·5-s − 2.64i·7-s − 25.2·9-s − 3.01i·11-s + 9.78·13-s − 33.8i·15-s − 11.6·17-s + 25.5i·19-s + 15.4·21-s − 26.1i·23-s + 8.42·25-s − 95.4i·27-s + 1.56·29-s + 12.0i·31-s + ⋯ |
| L(s) = 1 | + 1.95i·3-s − 1.15·5-s − 0.377i·7-s − 2.81·9-s − 0.274i·11-s + 0.752·13-s − 2.25i·15-s − 0.682·17-s + 1.34i·19-s + 0.737·21-s − 1.13i·23-s + 0.337·25-s − 3.53i·27-s + 0.0539·29-s + 0.387i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.165724 - 0.400094i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.165724 - 0.400094i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + 2.64iT \) |
| good | 3 | \( 1 - 5.85iT - 9T^{2} \) |
| 5 | \( 1 + 5.78T + 25T^{2} \) |
| 11 | \( 1 + 3.01iT - 121T^{2} \) |
| 13 | \( 1 - 9.78T + 169T^{2} \) |
| 17 | \( 1 + 11.6T + 289T^{2} \) |
| 19 | \( 1 - 25.5iT - 361T^{2} \) |
| 23 | \( 1 + 26.1iT - 529T^{2} \) |
| 29 | \( 1 - 1.56T + 841T^{2} \) |
| 31 | \( 1 - 12.0iT - 961T^{2} \) |
| 37 | \( 1 + 70.6T + 1.36e3T^{2} \) |
| 41 | \( 1 + 49.8T + 1.68e3T^{2} \) |
| 43 | \( 1 - 73.2iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 44.2iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 54.2T + 2.80e3T^{2} \) |
| 59 | \( 1 + 12.4iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 35.6T + 3.72e3T^{2} \) |
| 67 | \( 1 - 24.4iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 11.0iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 74.3T + 5.32e3T^{2} \) |
| 79 | \( 1 - 22.8iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 48.1iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 67.4T + 7.92e3T^{2} \) |
| 97 | \( 1 + 7.75T + 9.40e3T^{2} \) |
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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
−12.29986700929205627204413473391, −11.28741046849696635312037752866, −10.74179106710631240128089140198, −9.890298197147183689324568885653, −8.686873668473789907842523660365, −8.116982743974330659796586445834, −6.32017512748308473546800351418, −4.94385049692587675553021475238, −4.01417257710490480911557626573, −3.32536055898489011179326849833,
0.22975822131839937082282203411, 1.92057028481631718850800641211, 3.41450533786963782326992307115, 5.33306535825978581483637634575, 6.67906752233349378235312101423, 7.26709286428471378469727880771, 8.276241446320759775154473173671, 8.925886975818080179900072348699, 11.00414167779311780641214216954, 11.71181559042772851237990714130
