| L(s) = 1 | + i·2-s − 4-s − 0.366·5-s + (−1.91 − 1.82i)7-s − i·8-s − 0.366i·10-s + 0.669i·11-s − 1.00i·13-s + (1.82 − 1.91i)14-s + 16-s + 4.98·17-s + 6.35i·19-s + 0.366·20-s − 0.669·22-s + 7.69i·23-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.5·4-s − 0.163·5-s + (−0.723 − 0.690i)7-s − 0.353i·8-s − 0.115i·10-s + 0.201i·11-s − 0.277i·13-s + (0.488 − 0.511i)14-s + 0.250·16-s + 1.21·17-s + 1.45i·19-s + 0.0819·20-s − 0.142·22-s + 1.60i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.723 - 0.690i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.723 - 0.690i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9277459545\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9277459545\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 + (1.91 + 1.82i)T \) |
| good | 5 | \( 1 + 0.366T + 5T^{2} \) |
| 11 | \( 1 - 0.669iT - 11T^{2} \) |
| 13 | \( 1 + 1.00iT - 13T^{2} \) |
| 17 | \( 1 - 4.98T + 17T^{2} \) |
| 19 | \( 1 - 6.35iT - 19T^{2} \) |
| 23 | \( 1 - 7.69iT - 23T^{2} \) |
| 29 | \( 1 - 1.82iT - 29T^{2} \) |
| 31 | \( 1 - 6.32iT - 31T^{2} \) |
| 37 | \( 1 + 5.16T + 37T^{2} \) |
| 41 | \( 1 + 4.31T + 41T^{2} \) |
| 43 | \( 1 + 4.49T + 43T^{2} \) |
| 47 | \( 1 - 8.32T + 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 - 8.72T + 59T^{2} \) |
| 61 | \( 1 - 4.95iT - 61T^{2} \) |
| 67 | \( 1 + 10.8T + 67T^{2} \) |
| 71 | \( 1 - 5.49iT - 71T^{2} \) |
| 73 | \( 1 + 4.07iT - 73T^{2} \) |
| 79 | \( 1 - 8.35T + 79T^{2} \) |
| 83 | \( 1 - 17.0T + 83T^{2} \) |
| 89 | \( 1 + 10.7T + 89T^{2} \) |
| 97 | \( 1 - 17.2iT - 97T^{2} \) |
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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
−10.04256140626930927165338415456, −9.372967855399432305068102767779, −8.253770989301396814810707892569, −7.57678975955886833669775974254, −6.94089488242300955064036962715, −5.87870187244537574728117455268, −5.25645014704911825755620641400, −3.85581480945914072386900039745, −3.37026224945291197688019600857, −1.38064709910352589139376749827,
0.42966886586169784389658448176, 2.18901191694688624637248884170, 3.06709617198236173142120956172, 4.06224386376477561422142052662, 5.14040506273816847586818895426, 6.04509847550219971132203555029, 6.96503694229837253560068118662, 8.090202530385070902681726938405, 8.856237958220302688771120722581, 9.583831088573363849179297115065
