| L(s) = 1 | + (0.996 + 0.0871i)2-s + (0.984 + 0.173i)4-s + (−1.32 + 1.80i)5-s + (−2.99 − 2.09i)7-s + (0.965 + 0.258i)8-s + (−1.47 + 1.67i)10-s + (−1.72 + 4.75i)11-s + (0.119 + 1.36i)13-s + (−2.80 − 2.35i)14-s + (0.939 + 0.342i)16-s + (−2.52 + 0.677i)17-s + (−1.41 + 0.814i)19-s + (−1.61 + 1.54i)20-s + (−2.13 + 4.58i)22-s + (2.26 + 3.22i)23-s + ⋯ |
| L(s) = 1 | + (0.704 + 0.0616i)2-s + (0.492 + 0.0868i)4-s + (−0.592 + 0.805i)5-s + (−1.13 − 0.792i)7-s + (0.341 + 0.0915i)8-s + (−0.467 + 0.530i)10-s + (−0.521 + 1.43i)11-s + (0.0330 + 0.377i)13-s + (−0.748 − 0.628i)14-s + (0.234 + 0.0855i)16-s + (−0.612 + 0.164i)17-s + (−0.323 + 0.186i)19-s + (−0.361 + 0.345i)20-s + (−0.455 + 0.977i)22-s + (0.471 + 0.673i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.721 - 0.692i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.721 - 0.692i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.400277 + 0.995366i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.400277 + 0.995366i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.996 - 0.0871i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (1.32 - 1.80i)T \) |
| good | 7 | \( 1 + (2.99 + 2.09i)T + (2.39 + 6.57i)T^{2} \) |
| 11 | \( 1 + (1.72 - 4.75i)T + (-8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (-0.119 - 1.36i)T + (-12.8 + 2.25i)T^{2} \) |
| 17 | \( 1 + (2.52 - 0.677i)T + (14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (1.41 - 0.814i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.26 - 3.22i)T + (-7.86 + 21.6i)T^{2} \) |
| 29 | \( 1 + (7.81 - 6.55i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (0.523 - 2.96i)T + (-29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (-0.303 - 1.13i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (0.511 - 0.609i)T + (-7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (1.45 + 3.12i)T + (-27.6 + 32.9i)T^{2} \) |
| 47 | \( 1 + (-6.08 + 8.69i)T + (-16.0 - 44.1i)T^{2} \) |
| 53 | \( 1 + (1.71 - 1.71i)T - 53iT^{2} \) |
| 59 | \( 1 + (-12.1 + 4.42i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (-1.13 - 6.45i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-5.38 + 0.471i)T + (65.9 - 11.6i)T^{2} \) |
| 71 | \( 1 + (12.3 + 7.13i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-0.898 + 3.35i)T + (-63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (8.03 + 9.57i)T + (-13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (0.797 - 9.11i)T + (-81.7 - 14.4i)T^{2} \) |
| 89 | \( 1 + (-7.31 - 12.6i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (3.03 - 1.41i)T + (62.3 - 74.3i)T^{2} \) |
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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
−10.52873483489790248809551815171, −10.04649388160096547180947647457, −8.936618058497072674806535779646, −7.46295671937263393312600134320, −7.12674065283965948871107181720, −6.43266039661828988948816778798, −5.12601773099108968112375710585, −4.02820913686291812030951731196, −3.40060418903334010095903758119, −2.14107936487811704035877653487,
0.39168442580688729079543079846, 2.51468282629833306805799665304, 3.42663169852314608719088920529, 4.43697816840021119158188096615, 5.63885828520349209602492894057, 6.04373011536500388615503588614, 7.28072319503849424223384703681, 8.345067141094187554666339502807, 8.963770718433691167907008476906, 9.905285948121463204409828143436
