L(s) = 1 | + (−1 − 1.73i)2-s + (−1.99 + 3.46i)4-s + (−3 − 5.19i)5-s + 7.99·8-s + (−6 + 10.3i)10-s + (−15 + 25.9i)11-s + 2·13-s + (−8 − 13.8i)16-s + (−33 + 57.1i)17-s + (26 + 45.0i)19-s + 24·20-s + 60·22-s + (−57 − 98.7i)23-s + (44.5 − 77.0i)25-s + (−2 − 3.46i)26-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.268 − 0.464i)5-s + 0.353·8-s + (−0.189 + 0.328i)10-s + (−0.411 + 0.712i)11-s + 0.0426·13-s + (−0.125 − 0.216i)16-s + (−0.470 + 0.815i)17-s + (0.313 + 0.543i)19-s + 0.268·20-s + 0.581·22-s + (−0.516 − 0.895i)23-s + (0.355 − 0.616i)25-s + (−0.0150 − 0.0261i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.386 + 0.922i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.053361001\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.053361001\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1 + 1.73i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (3 + 5.19i)T + (-62.5 + 108. i)T^{2} \) |
| 11 | \( 1 + (15 - 25.9i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 - 2T + 2.19e3T^{2} \) |
| 17 | \( 1 + (33 - 57.1i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-26 - 45.0i)T + (-3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (57 + 98.7i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 - 72T + 2.43e4T^{2} \) |
| 31 | \( 1 + (-98 + 169. i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-143 - 247. i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + 378T + 6.89e4T^{2} \) |
| 43 | \( 1 - 164T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-114 - 197. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-174 + 301. i)T + (-7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-174 + 301. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-53 - 91.7i)T + (-1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (298 - 516. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 - 630T + 3.57e5T^{2} \) |
| 73 | \( 1 + (-521 + 902. i)T + (-1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-44 - 76.2i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + 1.44e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + (687 + 1.18e3i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + 34T + 9.12e5T^{2} \) |
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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
−9.740992415260192216760967640354, −8.421329719653034666935414173357, −8.240383328163199529541607929321, −7.04495507028091005635289317297, −6.01771529573056678937269654869, −4.72738928702710004218330272263, −4.10960330366790843041070634525, −2.78522023454477561243176007189, −1.71707677918271716372354227296, −0.39633018111258639945708327238,
0.908403316575332959191683866485, 2.56448878178511533615433797122, 3.64499683281847913923765425896, 4.92811501049603110385809074857, 5.71739176850383292411010324277, 6.79882935808722504087402662393, 7.36776001539196694753564458443, 8.303940316831252033972606853681, 9.057431468015248125979292916824, 9.902790238805984039776109117079