L(s) = 1 | + (0.779 − 1.54i)3-s + (0.268 + 2.21i)5-s + 2.49·7-s + (−1.78 − 2.41i)9-s + 3.17·11-s + 5.31i·13-s + (3.64 + 1.31i)15-s − 2.70i·17-s + 8.13i·19-s + (1.94 − 3.86i)21-s + (4.12 − 2.44i)23-s + (−4.85 + 1.19i)25-s + (−5.12 + 0.877i)27-s + 2.82i·29-s − 6.28·31-s + ⋯ |
L(s) = 1 | + (0.450 − 0.892i)3-s + (0.119 + 0.992i)5-s + 0.944·7-s + (−0.594 − 0.804i)9-s + 0.958·11-s + 1.47i·13-s + (0.940 + 0.339i)15-s − 0.655i·17-s + 1.86i·19-s + (0.425 − 0.843i)21-s + (0.859 − 0.510i)23-s + (−0.971 + 0.238i)25-s + (−0.985 + 0.168i)27-s + 0.524i·29-s − 1.12·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 - 0.187i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.982 - 0.187i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.277525467\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.277525467\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.779 + 1.54i)T \) |
| 5 | \( 1 + (-0.268 - 2.21i)T \) |
| 23 | \( 1 + (-4.12 + 2.44i)T \) |
good | 7 | \( 1 - 2.49T + 7T^{2} \) |
| 11 | \( 1 - 3.17T + 11T^{2} \) |
| 13 | \( 1 - 5.31iT - 13T^{2} \) |
| 17 | \( 1 + 2.70iT - 17T^{2} \) |
| 19 | \( 1 - 8.13iT - 19T^{2} \) |
| 29 | \( 1 - 2.82iT - 29T^{2} \) |
| 31 | \( 1 + 6.28T + 31T^{2} \) |
| 37 | \( 1 - 3.82T + 37T^{2} \) |
| 41 | \( 1 + 3.17iT - 41T^{2} \) |
| 43 | \( 1 - 6.93T + 43T^{2} \) |
| 47 | \( 1 - 8.10T + 47T^{2} \) |
| 53 | \( 1 + 3.79iT - 53T^{2} \) |
| 59 | \( 1 - 11.2iT - 59T^{2} \) |
| 61 | \( 1 + 0.443iT - 61T^{2} \) |
| 67 | \( 1 + 6.68T + 67T^{2} \) |
| 71 | \( 1 + 14.4iT - 71T^{2} \) |
| 73 | \( 1 + 2.80iT - 73T^{2} \) |
| 79 | \( 1 + 6.08iT - 79T^{2} \) |
| 83 | \( 1 - 8.29iT - 83T^{2} \) |
| 89 | \( 1 - 9.80T + 89T^{2} \) |
| 97 | \( 1 - 16.6T + 97T^{2} \) |
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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
−9.299350792507749050072924886703, −8.898413822452752274979874067348, −7.75489090420807224883969674383, −7.23784864106524919226708325469, −6.49616636106170531663165096696, −5.73040894210443619419590792100, −4.30505608573284514535136580634, −3.43212551173657065292906082070, −2.18934738022653364157861843737, −1.45570511663442082348032853283,
1.00092736931986932886076514095, 2.40474500640577447688270042715, 3.63057643331641773868790038064, 4.55153319767701093437089344722, 5.14887104573350565943173144440, 5.94516988035533168383811135878, 7.43718232944490330983876644552, 8.117354449334830214418589043060, 8.973224776162388471966023915865, 9.243222784137207523385524418109