| L(s) = 1 | + 1.82i·2-s + 9.90i·3-s + 28.6·4-s − 18.0·6-s − 162. i·7-s + 110. i·8-s + 144.·9-s + 146.·11-s + 283. i·12-s − 1.07e3i·13-s + 295.·14-s + 716.·16-s − 1.60e3i·17-s + 263. i·18-s − 851.·19-s + ⋯ |
| L(s) = 1 | + 0.321i·2-s + 0.635i·3-s + 0.896·4-s − 0.204·6-s − 1.25i·7-s + 0.610i·8-s + 0.596·9-s + 0.366·11-s + 0.569i·12-s − 1.75i·13-s + 0.402·14-s + 0.700·16-s − 1.34i·17-s + 0.192i·18-s − 0.540·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.515038465\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.515038465\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| good | 2 | \( 1 - 1.82iT - 32T^{2} \) |
| 3 | \( 1 - 9.90iT - 243T^{2} \) |
| 7 | \( 1 + 162. iT - 1.68e4T^{2} \) |
| 11 | \( 1 - 146.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 1.07e3iT - 3.71e5T^{2} \) |
| 17 | \( 1 + 1.60e3iT - 1.41e6T^{2} \) |
| 19 | \( 1 + 851.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 2.22e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 + 2.87e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 7.97e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 9.11e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 - 1.98e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 3.44e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 1.18e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 - 2.21e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 1.62e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.86e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 6.56e3iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 3.30e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 3.22e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 + 1.28e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 4.23e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 - 3.37e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 5.34e4iT - 8.58e9T^{2} \) |
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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
−12.46596089499023418705221999482, −11.10089418524813385696380727328, −10.49688296559750736446271214824, −9.535384129981195105906453406814, −7.76683599535067334372953371956, −7.18678971344068599991375312946, −5.76824762451310741147809940507, −4.36865088076717358188999801884, −3.00686689345758670634978635904, −0.980870100728567856927147194776,
1.54975967539549078359547085720, 2.36591848535486503755786160366, 4.20119822449468753183823320535, 6.20212446551291390173505884687, 6.70496651260276594309999942727, 8.137813035468766820640363430477, 9.298787887943721809708124862681, 10.54571476389598444276637742183, 11.77087762299457982922719530916, 12.21139440871023284312753698979
