| L(s) = 1 | − 2-s + i·3-s + 4-s − 2i·5-s − i·6-s − 8-s − 9-s + 2i·10-s + 4i·11-s + i·12-s − 2·13-s + 2·15-s + 16-s + 18-s − 4·19-s − 2i·20-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577i·3-s + 0.5·4-s − 0.894i·5-s − 0.408i·6-s − 0.353·8-s − 0.333·9-s + 0.632i·10-s + 1.20i·11-s + 0.288i·12-s − 0.554·13-s + 0.516·15-s + 0.250·16-s + 0.235·18-s − 0.917·19-s − 0.447i·20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1734 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.970 + 0.242i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1734 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.970 + 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 - iT \) |
| 17 | \( 1 \) |
| good | 5 | \( 1 + 2iT - 5T^{2} \) |
| 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 - 4iT - 11T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + 10iT - 29T^{2} \) |
| 31 | \( 1 - 8iT - 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 + 10iT - 41T^{2} \) |
| 43 | \( 1 + 12T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 + 12T + 59T^{2} \) |
| 61 | \( 1 - 10iT - 61T^{2} \) |
| 67 | \( 1 + 12T + 67T^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 - 10iT - 73T^{2} \) |
| 79 | \( 1 - 8iT - 79T^{2} \) |
| 83 | \( 1 + 4T + 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 + 14iT - 97T^{2} \) |
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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
−8.933761392799839826898100758992, −8.433760468622332108854992008419, −7.48720674231698928069959600385, −6.72187031728737423164665189200, −5.62162483568493530275223343667, −4.74903973721194273586276255785, −4.10371494679489060674602657297, −2.66795200858653271220835373618, −1.61730430388947983317664254113, 0,
1.53145053578433420290544987660, 2.73718349619107287341725545787, 3.37270404614000534435414361942, 4.88089984248490371332633681887, 6.09997107027575999209746548857, 6.53071028041173935888121967469, 7.36640210656233403252744450253, 8.099035156557818667115623743402, 8.780335538845723747815883561169
