L(s) = 1 | + (−0.5 + 0.866i)2-s + 3-s + (−0.499 − 0.866i)4-s + (−0.5 + 0.866i)6-s + 0.999·8-s + 9-s + (−1 − 1.73i)11-s + (−0.499 − 0.866i)12-s + (−0.5 + 0.866i)16-s + (0.5 + 0.866i)17-s + (−0.5 + 0.866i)18-s + 19-s + 1.99·22-s + 0.999·24-s + 25-s + ⋯ |
L(s) = 1 | + (−0.5 + 0.866i)2-s + 3-s + (−0.499 − 0.866i)4-s + (−0.5 + 0.866i)6-s + 0.999·8-s + 9-s + (−1 − 1.73i)11-s + (−0.499 − 0.866i)12-s + (−0.5 + 0.866i)16-s + (0.5 + 0.866i)17-s + (−0.5 + 0.866i)18-s + 19-s + 1.99·22-s + 0.999·24-s + 25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.884 - 0.466i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.884 - 0.466i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.145459403\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.145459403\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 3 | \( 1 - T \) |
| 19 | \( 1 - T \) |
good | 5 | \( 1 - T^{2} \) |
| 7 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + T + T^{2} \) |
| 43 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + T + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 73 | \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
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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
−9.576653552210071286089572194378, −8.822890583496170453687575602297, −8.192851587799892192257133027893, −7.72070293778092032737757934644, −6.74558511018088958535175811911, −5.78476422841888052494694663692, −5.02791190707556246349661063140, −3.73521511468091973331326751073, −2.82281648285893078076516027249, −1.23469924277622365506106348909,
1.52505675615170011341675871588, 2.58378302575382613485139916506, 3.25330572035040421517049450868, 4.50366156497486134580940444674, 5.08793541727939659086968727682, 7.05118191042878853508342862901, 7.45203618342923373147519079661, 8.209570800648944948529434388791, 9.094589215196535197811536862598, 9.917453027297138826619245402409