| L(s) = 1 | + (−0.483 − 1.32i)2-s + (2.95 + 0.505i)3-s + (−1.53 + 1.28i)4-s + (−5.47 + 6.52i)5-s + (−0.759 − 4.17i)6-s + (6.07 + 10.5i)7-s + (2.44 + 1.41i)8-s + (8.48 + 2.98i)9-s + (11.3 + 4.12i)10-s + (−6.93 − 4.00i)11-s + (−5.17 + 3.02i)12-s + (−1.68 − 9.56i)13-s + (11.0 − 13.1i)14-s + (−19.4 + 16.5i)15-s + (0.694 − 3.93i)16-s + (−4.97 − 13.6i)17-s + ⋯ |
| L(s) = 1 | + (−0.241 − 0.664i)2-s + (0.985 + 0.168i)3-s + (−0.383 + 0.321i)4-s + (−1.09 + 1.30i)5-s + (−0.126 − 0.695i)6-s + (0.867 + 1.50i)7-s + (0.306 + 0.176i)8-s + (0.943 + 0.331i)9-s + (1.13 + 0.412i)10-s + (−0.630 − 0.363i)11-s + (−0.431 + 0.252i)12-s + (−0.129 − 0.735i)13-s + (0.788 − 0.940i)14-s + (−1.29 + 1.10i)15-s + (0.0434 − 0.246i)16-s + (−0.292 − 0.803i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.784 - 0.620i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.784 - 0.620i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.32505 + 0.460754i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.32505 + 0.460754i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.483 + 1.32i)T \) |
| 3 | \( 1 + (-2.95 - 0.505i)T \) |
| 19 | \( 1 + (-18.9 + 1.92i)T \) |
| good | 5 | \( 1 + (5.47 - 6.52i)T + (-4.34 - 24.6i)T^{2} \) |
| 7 | \( 1 + (-6.07 - 10.5i)T + (-24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + (6.93 + 4.00i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (1.68 + 9.56i)T + (-158. + 57.8i)T^{2} \) |
| 17 | \( 1 + (4.97 + 13.6i)T + (-221. + 185. i)T^{2} \) |
| 23 | \( 1 + (-14.6 - 17.4i)T + (-91.8 + 520. i)T^{2} \) |
| 29 | \( 1 + (-0.608 + 1.67i)T + (-644. - 540. i)T^{2} \) |
| 31 | \( 1 + (-9.59 - 16.6i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + 4.28T + 1.36e3T^{2} \) |
| 41 | \( 1 + (-49.8 - 8.78i)T + (1.57e3 + 574. i)T^{2} \) |
| 43 | \( 1 + (44.9 + 37.7i)T + (321. + 1.82e3i)T^{2} \) |
| 47 | \( 1 + (-20.2 + 55.5i)T + (-1.69e3 - 1.41e3i)T^{2} \) |
| 53 | \( 1 + (-13.1 - 15.6i)T + (-487. + 2.76e3i)T^{2} \) |
| 59 | \( 1 + (2.80 + 7.71i)T + (-2.66e3 + 2.23e3i)T^{2} \) |
| 61 | \( 1 + (-2.11 + 1.77i)T + (646. - 3.66e3i)T^{2} \) |
| 67 | \( 1 + (25.1 + 9.17i)T + (3.43e3 + 2.88e3i)T^{2} \) |
| 71 | \( 1 + (-18.7 + 22.3i)T + (-875. - 4.96e3i)T^{2} \) |
| 73 | \( 1 + (-2.68 + 15.2i)T + (-5.00e3 - 1.82e3i)T^{2} \) |
| 79 | \( 1 + (-20.3 + 115. i)T + (-5.86e3 - 2.13e3i)T^{2} \) |
| 83 | \( 1 + (-24.5 + 14.1i)T + (3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + (80.5 - 14.2i)T + (7.44e3 - 2.70e3i)T^{2} \) |
| 97 | \( 1 + (-61.0 + 22.2i)T + (7.20e3 - 6.04e3i)T^{2} \) |
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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
−13.53555007639910663574035145279, −12.11580437985544296131567799644, −11.37518216204510582979161561351, −10.42918893193667625295635993193, −9.104231013306262984485425074826, −8.112221453324370658910231333235, −7.36185041338596478279385694339, −5.10929154275769940686139822663, −3.30720847294240457665883966165, −2.54936095619647493068133588595,
1.13460046910333519346428281680, 4.08696652486645092073881965199, 4.72592070838530962275727663405, 7.14548609919949219539627916239, 7.85722447606690796877853269152, 8.521617731503920959030290928613, 9.717198850895186892609346218535, 11.08395678457225527096805167637, 12.51914756222379510639783736081, 13.39320594809923607368551450894
