L(s) = 1 | − 1.73i·13-s − 1.73i·19-s + 25-s + 1.73i·31-s + 37-s + 43-s − 67-s − 1.73i·73-s − 79-s + 1.73i·103-s + 109-s + ⋯ |
L(s) = 1 | − 1.73i·13-s − 1.73i·19-s + 25-s + 1.73i·31-s + 37-s + 43-s − 67-s − 1.73i·73-s − 79-s + 1.73i·103-s + 109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 + 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 + 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.118390064\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.118390064\) |
\(L(1)\) |
|
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 | \( 1 - T^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( 1 + 1.73iT - T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 + 1.73iT - T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 - 1.73iT - T^{2} \) |
| 37 | \( 1 - T + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - T + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + T + T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + 1.73iT - T^{2} \) |
| 79 | \( 1 + T + T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.257921809794012541669276835668, −8.694103321015126369281764338354, −7.79251351648400939042685222173, −7.08962852102827203835123985182, −6.19813868057511161059301623153, −5.24536026167022527772398936435, −4.61772661579888641664611156362, −3.26571334090267834472061865824, −2.63217218719953035275204927225, −0.942561592259345294468296927486,
1.50487015914138607533452316670, 2.57510274776060613681995637119, 3.91280339917285512320307417121, 4.43807658020438802710731254167, 5.67759528043369136618398569354, 6.34061604149370236202950670022, 7.22425857804538120847758920325, 7.990575680104474115606634308158, 8.865857949801841562228746163395, 9.546241743492637231301743564817