| L(s) = 1 | + (−0.707 + 1.22i)2-s + (−2.72 − 1.25i)3-s + (−0.999 − 1.73i)4-s + 1.77·5-s + (3.46 − 2.44i)6-s + (10.9 − 6.34i)7-s + 2.82·8-s + (5.84 + 6.84i)9-s + (−1.25 + 2.17i)10-s + (5.17 − 8.96i)11-s + (0.548 + 5.97i)12-s + (−11.8 − 5.23i)13-s + 17.9i·14-s + (−4.84 − 2.23i)15-s + (−2.00 + 3.46i)16-s + (10.1 − 5.87i)17-s + ⋯ |
| L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.908 − 0.418i)3-s + (−0.249 − 0.433i)4-s + 0.355·5-s + (0.577 − 0.408i)6-s + (1.56 − 0.906i)7-s + 0.353·8-s + (0.649 + 0.760i)9-s + (−0.125 + 0.217i)10-s + (0.470 − 0.814i)11-s + (0.0457 + 0.497i)12-s + (−0.915 − 0.402i)13-s + 1.28i·14-s + (−0.323 − 0.148i)15-s + (−0.125 + 0.216i)16-s + (0.598 − 0.345i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.952 + 0.305i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.952 + 0.305i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.931755 - 0.145756i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.931755 - 0.145756i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.707 - 1.22i)T \) |
| 3 | \( 1 + (2.72 + 1.25i)T \) |
| 13 | \( 1 + (11.8 + 5.23i)T \) |
| good | 5 | \( 1 - 1.77T + 25T^{2} \) |
| 7 | \( 1 + (-10.9 + 6.34i)T + (24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (-5.17 + 8.96i)T + (-60.5 - 104. i)T^{2} \) |
| 17 | \( 1 + (-10.1 + 5.87i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (0.0496 - 0.0286i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (-27.0 - 15.6i)T + (264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + (26.2 + 15.1i)T + (420.5 + 728. i)T^{2} \) |
| 31 | \( 1 - 8.94iT - 961T^{2} \) |
| 37 | \( 1 + (-45.2 - 26.1i)T + (684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 + (26.8 - 46.5i)T + (-840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (-3.38 - 5.86i)T + (-924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 - 7.62T + 2.20e3T^{2} \) |
| 53 | \( 1 - 83.0iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (23.4 + 40.6i)T + (-1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-4.75 - 8.22i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (34.9 + 20.1i)T + (2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 + (5.96 + 10.3i)T + (-2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 - 41.6iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 52.4T + 6.24e3T^{2} \) |
| 83 | \( 1 + 46.5T + 6.88e3T^{2} \) |
| 89 | \( 1 + (67.4 - 116. i)T + (-3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (-62.8 + 36.2i)T + (4.70e3 - 8.14e3i)T^{2} \) |
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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
−14.15068365235642502416342952763, −13.30597642928125753904781999329, −11.68240987111043370759312142379, −10.95126137506808409883237730272, −9.749223717949211193756680025076, −8.003590914564982789434891141819, −7.26223501148494447679350457434, −5.76124863043833103822022902440, −4.70789761521290239924947962148, −1.20164022924834759949432205703,
1.86836301359651187547635559949, 4.46256080352633956839266053641, 5.51301891543385759291728067893, 7.34752066651836138137154444934, 8.918480177602393451890230843897, 9.904357657298497627860036770751, 11.10665396405750923323871405896, 11.86612847729029183128683817682, 12.65341129561971418504548160934, 14.51602971775406192502117103502
