| L(s) = 1 | + (1.93 + 0.340i)2-s + (0.120 − 0.143i)3-s + (1.73 + 0.632i)4-s + (−2.15 − 0.612i)5-s + (0.282 − 0.236i)6-s + (0.586 − 0.338i)7-s + (−0.257 − 0.148i)8-s + (0.514 + 2.91i)9-s + (−3.94 − 1.91i)10-s + (−1.42 + 2.46i)11-s + (0.300 − 0.173i)12-s + (−3.06 − 3.65i)13-s + (1.24 − 0.454i)14-s + (−0.347 + 0.235i)15-s + (−3.27 − 2.75i)16-s + (5.10 + 0.900i)17-s + ⋯ |
| L(s) = 1 | + (1.36 + 0.240i)2-s + (0.0696 − 0.0830i)3-s + (0.868 + 0.316i)4-s + (−0.961 − 0.274i)5-s + (0.115 − 0.0966i)6-s + (0.221 − 0.127i)7-s + (−0.0909 − 0.0525i)8-s + (0.171 + 0.973i)9-s + (−1.24 − 0.606i)10-s + (−0.428 + 0.741i)11-s + (0.0867 − 0.0500i)12-s + (−0.849 − 1.01i)13-s + (0.333 − 0.121i)14-s + (−0.0897 + 0.0607i)15-s + (−0.819 − 0.687i)16-s + (1.23 + 0.218i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 - 0.165i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.986 - 0.165i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.62385 + 0.135571i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.62385 + 0.135571i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (2.15 + 0.612i)T \) |
| 19 | \( 1 + (-3.00 + 3.15i)T \) |
| good | 2 | \( 1 + (-1.93 - 0.340i)T + (1.87 + 0.684i)T^{2} \) |
| 3 | \( 1 + (-0.120 + 0.143i)T + (-0.520 - 2.95i)T^{2} \) |
| 7 | \( 1 + (-0.586 + 0.338i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (1.42 - 2.46i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (3.06 + 3.65i)T + (-2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-5.10 - 0.900i)T + (15.9 + 5.81i)T^{2} \) |
| 23 | \( 1 + (0.359 - 0.987i)T + (-17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (-0.247 - 1.40i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (0.135 + 0.234i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 0.603iT - 37T^{2} \) |
| 41 | \( 1 + (-5.15 - 4.32i)T + (7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (1.92 + 5.28i)T + (-32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (7.77 - 1.37i)T + (44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + (2.35 - 6.47i)T + (-40.6 - 34.0i)T^{2} \) |
| 59 | \( 1 + (-1.75 + 9.96i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (-7.02 - 2.55i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (4.05 - 0.714i)T + (62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (-7.14 + 2.60i)T + (54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (-10.3 + 12.3i)T + (-12.6 - 71.8i)T^{2} \) |
| 79 | \( 1 + (11.3 + 9.54i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (12.2 - 7.09i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (6.31 - 5.29i)T + (15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (5.64 + 0.994i)T + (91.1 + 33.1i)T^{2} \) |
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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
−14.04037294545909106725279960902, −12.87595809304344458398096048370, −12.37928746037029949492324236685, −11.20403652054549304631509399105, −9.838512783388865030492879129260, −7.950738649690420080646046078751, −7.27130759086628394230326839719, −5.33415071205339467578992694390, −4.64316219871177274918185662267, −3.07346136880988881008683818349,
3.09964143925584336037078368913, 4.11872259833425170054916535551, 5.47700292752309601749973606821, 6.88456725951318445857883444575, 8.279471638115567049935578807608, 9.778231556799493172995306541994, 11.40895284134723795795985915290, 11.93904465079112556821611542726, 12.77248741244904539581482978267, 14.30382948198138804080346518446
