| L(s) = 1 | + (−0.973 + 0.433i)2-s + (−1.71 − 0.271i)3-s + (−0.578 + 0.642i)4-s + (−2.03 − 0.919i)5-s + (1.78 − 0.477i)6-s + (−0.798 + 1.38i)7-s + (0.943 − 2.90i)8-s + (2.85 + 0.927i)9-s + (2.38 + 0.0111i)10-s + (4.35 − 1.93i)11-s + (1.16 − 0.942i)12-s + (0.904 + 0.402i)13-s + (0.177 − 1.69i)14-s + (3.23 + 2.12i)15-s + (0.159 + 1.51i)16-s + (0.0927 − 0.285i)17-s + ⋯ |
| L(s) = 1 | + (−0.688 + 0.306i)2-s + (−0.987 − 0.156i)3-s + (−0.289 + 0.321i)4-s + (−0.911 − 0.410i)5-s + (0.727 − 0.194i)6-s + (−0.301 + 0.522i)7-s + (0.333 − 1.02i)8-s + (0.950 + 0.309i)9-s + (0.753 + 0.00351i)10-s + (1.31 − 0.584i)11-s + (0.336 − 0.272i)12-s + (0.250 + 0.111i)13-s + (0.0475 − 0.452i)14-s + (0.836 + 0.548i)15-s + (0.0397 + 0.378i)16-s + (0.0224 − 0.0692i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.952 + 0.304i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.952 + 0.304i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.459715 - 0.0717851i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.459715 - 0.0717851i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.71 + 0.271i)T \) |
| 5 | \( 1 + (2.03 + 0.919i)T \) |
| good | 2 | \( 1 + (0.973 - 0.433i)T + (1.33 - 1.48i)T^{2} \) |
| 7 | \( 1 + (0.798 - 1.38i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-4.35 + 1.93i)T + (7.36 - 8.17i)T^{2} \) |
| 13 | \( 1 + (-0.904 - 0.402i)T + (8.69 + 9.66i)T^{2} \) |
| 17 | \( 1 + (-0.0927 + 0.285i)T + (-13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (-0.468 + 1.44i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (-0.597 + 5.68i)T + (-22.4 - 4.78i)T^{2} \) |
| 29 | \( 1 + (-5.78 - 1.23i)T + (26.4 + 11.7i)T^{2} \) |
| 31 | \( 1 + (-2.67 + 0.567i)T + (28.3 - 12.6i)T^{2} \) |
| 37 | \( 1 + (-1.92 + 1.39i)T + (11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (10.2 + 4.58i)T + (27.4 + 30.4i)T^{2} \) |
| 43 | \( 1 + (-3.80 + 6.58i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (5.69 + 1.21i)T + (42.9 + 19.1i)T^{2} \) |
| 53 | \( 1 + (-0.598 - 1.84i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (-5.31 - 2.36i)T + (39.4 + 43.8i)T^{2} \) |
| 61 | \( 1 + (-12.1 + 5.41i)T + (40.8 - 45.3i)T^{2} \) |
| 67 | \( 1 + (-1.02 + 0.217i)T + (61.2 - 27.2i)T^{2} \) |
| 71 | \( 1 + (5.09 + 15.6i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-7.42 - 5.39i)T + (22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (10.9 + 2.32i)T + (72.1 + 32.1i)T^{2} \) |
| 83 | \( 1 + (-3.98 - 4.42i)T + (-8.67 + 82.5i)T^{2} \) |
| 89 | \( 1 + (-1.01 - 0.740i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (-1.87 - 0.398i)T + (88.6 + 39.4i)T^{2} \) |
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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
−12.12714576577851399485797933665, −11.41538153278236115742955249316, −10.21081152291811799020291907949, −8.993188606954056614539619336593, −8.413421972130351552568491028131, −7.11611287583904499532669760290, −6.34515047435676229079846332093, −4.78375557107051858355642978011, −3.71561132172364177607292549757, −0.73367888372546673377740372343,
1.13725111911766370006288466391, 3.78254400855199067746857388752, 4.80866723901749609273234631929, 6.29833926043664578444474321330, 7.22365952444668942968423211304, 8.454150418145359223976478656412, 9.784669683425448651085951879464, 10.21985572334239487206106615783, 11.48447915470915537883972229440, 11.67714944417611212032350752201
