L(s) = 1 | + (−0.988 − 3.87i)2-s + (−14.0 + 7.66i)4-s + 20.7·5-s − 5.38i·7-s + (43.5 + 46.8i)8-s + (−20.5 − 80.4i)10-s + 115. i·11-s + 207.·13-s + (−20.8 + 5.32i)14-s + (138. − 215. i)16-s + 383.·17-s − 618. i·19-s + (−291. + 159. i)20-s + (448. − 114. i)22-s − 82.9i·23-s + ⋯ |
L(s) = 1 | + (−0.247 − 0.968i)2-s + (−0.877 + 0.479i)4-s + 0.830·5-s − 0.109i·7-s + (0.681 + 0.732i)8-s + (−0.205 − 0.804i)10-s + 0.955i·11-s + 1.22·13-s + (−0.106 + 0.0271i)14-s + (0.540 − 0.841i)16-s + 1.32·17-s − 1.71i·19-s + (−0.728 + 0.397i)20-s + (0.925 − 0.236i)22-s − 0.156i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.479 + 0.877i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.479 + 0.877i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(1.47724 - 0.876697i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.47724 - 0.876697i\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.988 + 3.87i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 - 20.7T + 625T^{2} \) |
| 7 | \( 1 + 5.38iT - 2.40e3T^{2} \) |
| 11 | \( 1 - 115. iT - 1.46e4T^{2} \) |
| 13 | \( 1 - 207.T + 2.85e4T^{2} \) |
| 17 | \( 1 - 383.T + 8.35e4T^{2} \) |
| 19 | \( 1 + 618. iT - 1.30e5T^{2} \) |
| 23 | \( 1 + 82.9iT - 2.79e5T^{2} \) |
| 29 | \( 1 - 201.T + 7.07e5T^{2} \) |
| 31 | \( 1 + 196. iT - 9.23e5T^{2} \) |
| 37 | \( 1 - 340.T + 1.87e6T^{2} \) |
| 41 | \( 1 - 2.79e3T + 2.82e6T^{2} \) |
| 43 | \( 1 + 254. iT - 3.41e6T^{2} \) |
| 47 | \( 1 - 2.25e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 - 4.11e3T + 7.89e6T^{2} \) |
| 59 | \( 1 + 1.59e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 + 6.08e3T + 1.38e7T^{2} \) |
| 67 | \( 1 - 7.38e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 - 9.34e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 5.32e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + 5.97e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 + 1.00e4iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 1.36e4T + 6.27e7T^{2} \) |
| 97 | \( 1 - 2.17e3T + 8.85e7T^{2} \) |
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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
−12.83368198450549807436021418801, −11.68086805338332852069459412007, −10.61978066715580885881563597645, −9.720817531296385187660520919413, −8.837711013040690044201811450245, −7.41929192289830599347663685594, −5.72782063382163470430732033635, −4.27904658891333317186815482214, −2.64578806305633275587880393088, −1.15237511634133615653486080100,
1.22252485037696754735893871362, 3.71381587242563638378526043325, 5.68798500946321975032376518880, 6.05117872847249066240952518470, 7.73600903762575280856741038497, 8.682571667912603866932873797388, 9.791937983662076891037696752459, 10.71704570377874149971552564046, 12.34119402631932107719299090353, 13.66896181258778644641830673948