L(s) = 1 | − 0.765·5-s + 2i·11-s − 0.317i·13-s − 5.54·17-s − 3.69i·19-s + 3.17i·23-s − 4.41·25-s + 6.82i·29-s − 6.75i·31-s − 0.242·37-s + 2.74·41-s − 6.82·43-s + 11.9·47-s − 12.2i·53-s − 1.53i·55-s + ⋯ |
L(s) = 1 | − 0.342·5-s + 0.603i·11-s − 0.0879i·13-s − 1.34·17-s − 0.847i·19-s + 0.661i·23-s − 0.882·25-s + 1.26i·29-s − 1.21i·31-s − 0.0398·37-s + 0.428·41-s − 1.04·43-s + 1.74·47-s − 1.68i·53-s − 0.206i·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.896 + 0.442i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.896 + 0.442i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.363370023\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.363370023\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + 0.765T + 5T^{2} \) |
| 11 | \( 1 - 2iT - 11T^{2} \) |
| 13 | \( 1 + 0.317iT - 13T^{2} \) |
| 17 | \( 1 + 5.54T + 17T^{2} \) |
| 19 | \( 1 + 3.69iT - 19T^{2} \) |
| 23 | \( 1 - 3.17iT - 23T^{2} \) |
| 29 | \( 1 - 6.82iT - 29T^{2} \) |
| 31 | \( 1 + 6.75iT - 31T^{2} \) |
| 37 | \( 1 + 0.242T + 37T^{2} \) |
| 41 | \( 1 - 2.74T + 41T^{2} \) |
| 43 | \( 1 + 6.82T + 43T^{2} \) |
| 47 | \( 1 - 11.9T + 47T^{2} \) |
| 53 | \( 1 + 12.2iT - 53T^{2} \) |
| 59 | \( 1 - 13.2T + 59T^{2} \) |
| 61 | \( 1 - 3.56iT - 61T^{2} \) |
| 67 | \( 1 - 4.48T + 67T^{2} \) |
| 71 | \( 1 + 9.31iT - 71T^{2} \) |
| 73 | \( 1 - 11.8iT - 73T^{2} \) |
| 79 | \( 1 - 11.3T + 79T^{2} \) |
| 83 | \( 1 - 4.32T + 83T^{2} \) |
| 89 | \( 1 + 1.66T + 89T^{2} \) |
| 97 | \( 1 + 11.8iT - 97T^{2} \) |
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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
−7.87799284731825052248440663862, −7.06327574467328024582037198964, −6.74506522355782640884002251949, −5.71120362870700763611733309687, −5.03732980912410758310580039322, −4.24569896198141240346188516900, −3.65079408581319162283163072112, −2.54610498456801539280325423888, −1.86252699715047978992491251418, −0.48726836870764441151489964093,
0.68721712199752029433449252416, 1.95530772859399480074207728494, 2.73788075551882949494270885236, 3.80466248914284176579091936725, 4.23873796260130800583157286124, 5.17751901710621948508787162775, 5.98044781242516734999047483750, 6.54873842509669207651698364282, 7.30966217723995859459736373348, 8.094428789661900243946808959720