L(s) = 1 | − 1.84·5-s − 2i·11-s + 4.46i·13-s + 2.29·17-s − 1.53i·19-s − 8.82i·23-s − 1.58·25-s + 1.17i·29-s + 5.86i·31-s + 8.24·37-s + 11.8·41-s + 1.17·43-s − 8.02·47-s − 3.75i·53-s + 3.69i·55-s + ⋯ |
L(s) = 1 | − 0.826·5-s − 0.603i·11-s + 1.23i·13-s + 0.556·17-s − 0.351i·19-s − 1.84i·23-s − 0.317·25-s + 0.217i·29-s + 1.05i·31-s + 1.35·37-s + 1.85·41-s + 0.178·43-s − 1.17·47-s − 0.516i·53-s + 0.498i·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.860 + 0.508i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.860 + 0.508i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.344379571\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.344379571\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.84T + 5T^{2} \) |
| 11 | \( 1 + 2iT - 11T^{2} \) |
| 13 | \( 1 - 4.46iT - 13T^{2} \) |
| 17 | \( 1 - 2.29T + 17T^{2} \) |
| 19 | \( 1 + 1.53iT - 19T^{2} \) |
| 23 | \( 1 + 8.82iT - 23T^{2} \) |
| 29 | \( 1 - 1.17iT - 29T^{2} \) |
| 31 | \( 1 - 5.86iT - 31T^{2} \) |
| 37 | \( 1 - 8.24T + 37T^{2} \) |
| 41 | \( 1 - 11.8T + 41T^{2} \) |
| 43 | \( 1 - 1.17T + 43T^{2} \) |
| 47 | \( 1 + 8.02T + 47T^{2} \) |
| 53 | \( 1 + 3.75iT - 53T^{2} \) |
| 59 | \( 1 - 9.81T + 59T^{2} \) |
| 61 | \( 1 + 12.3iT - 61T^{2} \) |
| 67 | \( 1 - 12.4T + 67T^{2} \) |
| 71 | \( 1 + 13.3iT - 71T^{2} \) |
| 73 | \( 1 + 2.74iT - 73T^{2} \) |
| 79 | \( 1 - 11.3T + 79T^{2} \) |
| 83 | \( 1 - 10.4T + 83T^{2} \) |
| 89 | \( 1 + 14.4T + 89T^{2} \) |
| 97 | \( 1 - 2.74iT - 97T^{2} \) |
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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
−9.160549175964570283228003009489, −8.375419836597499739022617400301, −7.77616247254958757086689270662, −6.78349847947643574901816602334, −6.20674909417279214676383829796, −4.97194947451148259310408377545, −4.23477527593013707140037839779, −3.37279743824186146531828830380, −2.23331809833142850814003886467, −0.67520443543456329616929782731,
0.949516211440669714084407610682, 2.46605597469200169521236291768, 3.56488869578384213870549419969, 4.23383352685307655859618784176, 5.40540931249902720108902584277, 5.98934434509120405241661162118, 7.34124198902901756479013746340, 7.70887308301174859496874300826, 8.337149629047287482227228425420, 9.651460758930827215050605941255