| L(s) = 1 | − 2i·3-s + (1 + 2i)5-s + (3 + 3i)7-s − 9-s + (1 + i)11-s − 2·13-s + (4 − 2i)15-s + (1 + i)17-s + (−3 − 3i)19-s + (6 − 6i)21-s + (1 − i)23-s + (−3 + 4i)25-s − 4i·27-s + (7 − 7i)29-s − 2i·31-s + ⋯ |
| L(s) = 1 | − 1.15i·3-s + (0.447 + 0.894i)5-s + (1.13 + 1.13i)7-s − 0.333·9-s + (0.301 + 0.301i)11-s − 0.554·13-s + (1.03 − 0.516i)15-s + (0.242 + 0.242i)17-s + (−0.688 − 0.688i)19-s + (1.30 − 1.30i)21-s + (0.208 − 0.208i)23-s + (−0.600 + 0.800i)25-s − 0.769i·27-s + (1.29 − 1.29i)29-s − 0.359i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 + 0.160i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.987 + 0.160i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.55441 - 0.125303i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.55441 - 0.125303i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-1 - 2i)T \) |
| good | 3 | \( 1 + 2iT - 3T^{2} \) |
| 7 | \( 1 + (-3 - 3i)T + 7iT^{2} \) |
| 11 | \( 1 + (-1 - i)T + 11iT^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 + (-1 - i)T + 17iT^{2} \) |
| 19 | \( 1 + (3 + 3i)T + 19iT^{2} \) |
| 23 | \( 1 + (-1 + i)T - 23iT^{2} \) |
| 29 | \( 1 + (-7 + 7i)T - 29iT^{2} \) |
| 31 | \( 1 + 2iT - 31T^{2} \) |
| 37 | \( 1 + 6T + 37T^{2} \) |
| 41 | \( 1 + 4iT - 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 + (7 - 7i)T - 47iT^{2} \) |
| 53 | \( 1 - 8iT - 53T^{2} \) |
| 59 | \( 1 + (-3 + 3i)T - 59iT^{2} \) |
| 61 | \( 1 + (1 + i)T + 61iT^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + (3 + 3i)T + 73iT^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + 2iT - 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + (11 + 11i)T + 97iT^{2} \) |
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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
−11.81853148565904024045506306947, −10.85972639865483693286913601197, −9.753678489553911571767561693619, −8.570992365075063202755649707221, −7.71631869699299992646644216015, −6.76307353317260986551227096086, −5.95958106222581037048951865545, −4.68421099951534354003189335228, −2.61823717490455430736711609819, −1.80819253553579409390043188520,
1.44839225881263002157215345488, 3.66745985351084340687865058748, 4.70574979518464936507486512757, 5.18088978233152992064949977803, 6.83672439412938143272034418773, 8.123015815808663174689436146854, 8.885835115585751563821797233951, 10.03828948999061258209431281768, 10.44146808139289874391322627518, 11.49478479548073334636179847494
