| L(s) = 1 | + i·2-s + i·3-s − 4-s − 3.26i·5-s − 6-s − i·8-s − 9-s + 3.26·10-s + (3.11 + 1.14i)11-s − i·12-s + 5.12·13-s + 3.26·15-s + 16-s − 5.32·17-s − i·18-s − 4.27·19-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s + 0.577i·3-s − 0.5·4-s − 1.46i·5-s − 0.408·6-s − 0.353i·8-s − 0.333·9-s + 1.03·10-s + (0.939 + 0.343i)11-s − 0.288i·12-s + 1.42·13-s + 0.843·15-s + 0.250·16-s − 1.29·17-s − 0.235i·18-s − 0.980·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.934 - 0.354i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.934 - 0.354i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.811797892\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.811797892\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 - iT \) |
| 7 | \( 1 \) |
| 11 | \( 1 + (-3.11 - 1.14i)T \) |
| good | 5 | \( 1 + 3.26iT - 5T^{2} \) |
| 13 | \( 1 - 5.12T + 13T^{2} \) |
| 17 | \( 1 + 5.32T + 17T^{2} \) |
| 19 | \( 1 + 4.27T + 19T^{2} \) |
| 23 | \( 1 - 5.37T + 23T^{2} \) |
| 29 | \( 1 - 4.77iT - 29T^{2} \) |
| 31 | \( 1 - 6.46iT - 31T^{2} \) |
| 37 | \( 1 - 2.13T + 37T^{2} \) |
| 41 | \( 1 + 0.949T + 41T^{2} \) |
| 43 | \( 1 + 6.20iT - 43T^{2} \) |
| 47 | \( 1 + 12.0iT - 47T^{2} \) |
| 53 | \( 1 - 12.5T + 53T^{2} \) |
| 59 | \( 1 + 13.4iT - 59T^{2} \) |
| 61 | \( 1 - 5.34T + 61T^{2} \) |
| 67 | \( 1 - 6.19T + 67T^{2} \) |
| 71 | \( 1 + 10.8T + 71T^{2} \) |
| 73 | \( 1 - 13.2T + 73T^{2} \) |
| 79 | \( 1 - 9.16iT - 79T^{2} \) |
| 83 | \( 1 + 0.835T + 83T^{2} \) |
| 89 | \( 1 - 6.22iT - 89T^{2} \) |
| 97 | \( 1 + 0.624iT - 97T^{2} \) |
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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
−8.687258196296942430534591350219, −8.454233366629611802800991120272, −6.96694419389934659895606324898, −6.54546285780531738199133503871, −5.49426324801252442911738725182, −4.92123364002018236418510412718, −4.17610855668818072271143696765, −3.59550263912521903243439451272, −1.85528230593989528276360160258, −0.74239451148666151060875932193,
0.940995186793132486034605140703, 2.14294330107116581283409952859, 2.83312537944026296442007028890, 3.76217412695820217774095753642, 4.39042728206089580584827283352, 5.98526598869178432539838925105, 6.30847885480438546669776455588, 6.98593680308464152673092981376, 7.896845195818756565182789101020, 8.762846400444112259228879149443
