L(s) = 1 | + (1.07 + 1.36i)3-s − 2.57·5-s + (−0.703 + 2.91i)9-s − 1.65i·11-s − 5.71i·13-s + (−2.76 − 3.50i)15-s − 7.58·17-s − 2.99i·19-s + 0.287i·23-s + 1.65·25-s + (−4.72 + 2.16i)27-s + 2.05i·29-s − 6.01i·31-s + (2.25 − 1.77i)33-s + 1.75·37-s + ⋯ |
L(s) = 1 | + (0.618 + 0.785i)3-s − 1.15·5-s + (−0.234 + 0.972i)9-s − 0.498i·11-s − 1.58i·13-s + (−0.713 − 0.906i)15-s − 1.83·17-s − 0.686i·19-s + 0.0600i·23-s + 0.330·25-s + (−0.908 + 0.417i)27-s + 0.382i·29-s − 1.08i·31-s + (0.391 − 0.308i)33-s + 0.288·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.188 + 0.982i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.188 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6247549316\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6247549316\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.07 - 1.36i)T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + 2.57T + 5T^{2} \) |
| 11 | \( 1 + 1.65iT - 11T^{2} \) |
| 13 | \( 1 + 5.71iT - 13T^{2} \) |
| 17 | \( 1 + 7.58T + 17T^{2} \) |
| 19 | \( 1 + 2.99iT - 19T^{2} \) |
| 23 | \( 1 - 0.287iT - 23T^{2} \) |
| 29 | \( 1 - 2.05iT - 29T^{2} \) |
| 31 | \( 1 + 6.01iT - 31T^{2} \) |
| 37 | \( 1 - 1.75T + 37T^{2} \) |
| 41 | \( 1 - 4.28T + 41T^{2} \) |
| 43 | \( 1 - 2.46T + 43T^{2} \) |
| 47 | \( 1 - 0.373T + 47T^{2} \) |
| 53 | \( 1 + 7.77iT - 53T^{2} \) |
| 59 | \( 1 + 9.79T + 59T^{2} \) |
| 61 | \( 1 + 1.02iT - 61T^{2} \) |
| 67 | \( 1 - 2.36T + 67T^{2} \) |
| 71 | \( 1 + 15.6iT - 71T^{2} \) |
| 73 | \( 1 + 3.81iT - 73T^{2} \) |
| 79 | \( 1 + 9.12T + 79T^{2} \) |
| 83 | \( 1 + 6.65T + 83T^{2} \) |
| 89 | \( 1 + 14.5T + 89T^{2} \) |
| 97 | \( 1 - 4.43iT - 97T^{2} \) |
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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
−9.397747012681930927191249205203, −8.692045822300714637392825865752, −8.005733207378918220837582853470, −7.39314817936096859000370122362, −6.12368361621359903462572300620, −4.99361717895032256698741831679, −4.22337898626155630214626649521, −3.37370233133169181429097556070, −2.48519886739250895073206046087, −0.24084347121882866002917069710,
1.61884371466647499732910251965, 2.67612078872750795201390725916, 4.02648415121128944824047003420, 4.42344652270869749479878327006, 6.10961587235294539382506171479, 6.96960777716751380175310718133, 7.40248677821904675690075110533, 8.436584293990812896055045213769, 8.908879599336817414362622730738, 9.785249989056849364817876488702