L(s) = 1 | + (−0.866 + 0.5i)2-s − 1.73·3-s + (−0.500 + 0.866i)4-s + (1.86 − 1.23i)5-s + (1.49 − 0.866i)6-s + (−0.866 + 0.5i)7-s − 3i·8-s + 2.99·9-s + (−1 + 2i)10-s + (1.5 + 2.59i)11-s + (0.866 − 1.49i)12-s + (0.866 + 0.5i)13-s + (0.499 − 0.866i)14-s + (−3.23 + 2.13i)15-s + (0.500 + 0.866i)16-s + 7i·17-s + ⋯ |
L(s) = 1 | + (−0.612 + 0.353i)2-s − 1.00·3-s + (−0.250 + 0.433i)4-s + (0.834 − 0.550i)5-s + (0.612 − 0.353i)6-s + (−0.327 + 0.188i)7-s − 1.06i·8-s + 0.999·9-s + (−0.316 + 0.632i)10-s + (0.452 + 0.783i)11-s + (0.250 − 0.433i)12-s + (0.240 + 0.138i)13-s + (0.133 − 0.231i)14-s + (−0.834 + 0.550i)15-s + (0.125 + 0.216i)16-s + 1.69i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.232 - 0.972i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.232 - 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.381264 + 0.483068i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.381264 + 0.483068i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + 1.73T \) |
| 5 | \( 1 + (-1.86 + 1.23i)T \) |
| 7 | \( 1 + (0.866 - 0.5i)T \) |
good | 2 | \( 1 + (0.866 - 0.5i)T + (1 - 1.73i)T^{2} \) |
| 11 | \( 1 + (-1.5 - 2.59i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.866 - 0.5i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 - 7iT - 17T^{2} \) |
| 19 | \( 1 + 6T + 19T^{2} \) |
| 23 | \( 1 + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-5 - 8.66i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (4 - 6.92i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 8iT - 37T^{2} \) |
| 41 | \( 1 + (1 - 1.73i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-3.46 + 2i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-7.79 + 4.5i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 + (-2 + 3.46i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (5 + 8.66i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.46 - 2i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 5T + 71T^{2} \) |
| 73 | \( 1 - 3iT - 73T^{2} \) |
| 79 | \( 1 + (-3.5 - 6.06i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (4.33 - 2.5i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 18T + 89T^{2} \) |
| 97 | \( 1 + (4.33 - 2.5i)T + (48.5 - 84.0i)T^{2} \) |
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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
−12.36533305466716200343859333112, −10.65968226960591735998604050243, −10.08572923872973721075690474022, −9.041367744333648203797976083333, −8.357981900533143131951681217761, −6.81884260522802219274664819798, −6.35333956850444061378629582218, −5.01148511360456842377543932484, −3.93387277418203653643475218529, −1.55482727834239999611678247348,
0.65912522498567746488438829373, 2.38324604528608972742369904198, 4.35820986211682045510658817944, 5.73753709263598194581812969663, 6.20464139653574160431434222451, 7.44646811626751569166101267025, 9.019995820868694918559785398913, 9.651184422703226285169379244333, 10.57836344672918235032261326426, 11.05516294274860558872055561476