| L(s) = 1 | − 1.08e10i·2-s + (3.60e15 + 4.22e15i)3-s − 4.36e19·4-s + 5.11e22i·5-s + (4.58e25 − 3.90e25i)6-s − 1.55e27·7-s − 3.26e29i·8-s + (−4.87e30 + 3.05e31i)9-s + 5.53e32·10-s − 2.60e33i·11-s + (−1.57e35 − 1.84e35i)12-s − 3.62e36·13-s + 1.68e37i·14-s + (−2.16e38 + 1.84e38i)15-s − 6.76e39·16-s + 5.08e39i·17-s + ⋯ |
| L(s) = 1 | − 1.26i·2-s + (0.648 + 0.760i)3-s − 0.590·4-s + 0.439i·5-s + (0.959 − 0.818i)6-s − 0.201·7-s − 0.515i·8-s + (−0.157 + 0.987i)9-s + 0.553·10-s − 0.112i·11-s + (−0.383 − 0.449i)12-s − 0.630·13-s + 0.253i·14-s + (−0.334 + 0.284i)15-s − 1.24·16-s + 0.126i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.648 - 0.760i)\, \overline{\Lambda}(67-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+33) \, L(s)\cr =\mathstrut & (-0.648 - 0.760i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{67}{2})\) |
\(\approx\) |
\(0.1619217215\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1619217215\) |
| \(L(34)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-3.60e15 - 4.22e15i)T \) |
| good | 2 | \( 1 + 1.08e10iT - 7.37e19T^{2} \) |
| 5 | \( 1 - 5.11e22iT - 1.35e46T^{2} \) |
| 7 | \( 1 + 1.55e27T + 5.97e55T^{2} \) |
| 11 | \( 1 + 2.60e33iT - 5.39e68T^{2} \) |
| 13 | \( 1 + 3.62e36T + 3.31e73T^{2} \) |
| 17 | \( 1 - 5.08e39iT - 1.62e81T^{2} \) |
| 19 | \( 1 - 9.86e41T + 2.49e84T^{2} \) |
| 23 | \( 1 + 4.39e44iT - 7.48e89T^{2} \) |
| 29 | \( 1 - 2.28e48iT - 3.29e96T^{2} \) |
| 31 | \( 1 - 7.63e47T + 2.69e98T^{2} \) |
| 37 | \( 1 + 6.54e51T + 3.17e103T^{2} \) |
| 41 | \( 1 + 1.75e53iT - 2.77e106T^{2} \) |
| 43 | \( 1 + 1.40e54T + 6.44e107T^{2} \) |
| 47 | \( 1 + 2.43e55iT - 2.28e110T^{2} \) |
| 53 | \( 1 - 7.50e56iT - 6.34e113T^{2} \) |
| 59 | \( 1 + 8.16e56iT - 7.52e116T^{2} \) |
| 61 | \( 1 + 1.29e59T + 6.78e117T^{2} \) |
| 67 | \( 1 + 2.71e60T + 3.31e120T^{2} \) |
| 71 | \( 1 - 1.32e60iT - 1.52e122T^{2} \) |
| 73 | \( 1 - 1.70e61T + 9.53e122T^{2} \) |
| 79 | \( 1 + 1.49e62T + 1.75e125T^{2} \) |
| 83 | \( 1 + 3.30e62iT - 4.56e126T^{2} \) |
| 89 | \( 1 - 3.08e64iT - 4.56e128T^{2} \) |
| 97 | \( 1 + 3.78e65T + 1.33e131T^{2} \) |
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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
−13.72882964126091754858894045940, −12.25693232659841130845369187552, −10.83046579612942929402913987558, −10.04316329938050569551089564488, −8.836064315559589150450698166012, −7.03494953636808082333602637308, −4.93050864037357792230154300749, −3.53150022291039911794681400760, −2.83140461455909066113493543291, −1.65451080083781275407122887397,
0.03025561516265132057845197217, 1.57479667767063795344499034690, 2.97782129547733745266993173531, 4.85531442029227702889546376919, 6.23559206952937526869963503773, 7.30622785442626120064299970640, 8.242144918068558159538562013412, 9.468891810408334700712335132464, 11.82659154445516183375191751205, 13.25284777433138590321878683822
