| L(s) = 1 | + (0.707 − 1.22i)2-s + (2.45 + 1.73i)3-s + (−0.999 − 1.73i)4-s − 1.77·5-s + (3.85 − 1.77i)6-s + (10.9 − 6.34i)7-s − 2.82·8-s + (3.00 + 8.48i)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 + (−4.35 − 3.07i)15-s + (−2.00 + 3.46i)16-s + (−10.1 + 5.87i)17-s + ⋯ |
| L(s) = 1 | + (0.353 − 0.612i)2-s + (0.816 + 0.577i)3-s + (−0.249 − 0.433i)4-s − 0.355·5-s + (0.642 − 0.296i)6-s + (1.56 − 0.906i)7-s − 0.353·8-s + (0.333 + 0.942i)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 + (−0.290 − 0.205i)15-s + (−0.125 + 0.216i)16-s + (−0.598 + 0.345i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.880 + 0.473i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.880 + 0.473i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.77209 - 0.446283i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.77209 - 0.446283i\) |
| \(L(2)\) |
|
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 | \( 1 + (-2.45 - 1.73i)T \) |
| 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
−14.29802423950338608254721596456, −13.17018158630539163281377314361, −11.86380982915991394309022049617, −10.66458638231312578220050967299, −9.993736718151416317293226855765, −8.350332547675198178645023285000, −7.50587002630621198406292363645, −4.88972825760680133211706780801, −4.15306678663458166223741497572, −2.17847706373380927371728011403,
2.40233276619562336035941543070, 4.43084278255808772026978100547, 5.93612674187724955830505510419, 7.67023780642092918845173249553, 8.162353324821450087096695588173, 9.338829633698168621998709757721, 11.44438807971769661937610045083, 12.16491247781093838823601626723, 13.55059591149396321142239805156, 14.33168700331378888496600564753
