| L(s) = 1 | + (−6.55 − 3.78i)5-s + (−4.55 − 7.89i)7-s + (0.383 − 0.221i)11-s + (5.55 − 9.62i)13-s + 8.01i·17-s − 8.11·19-s + (−20.4 − 11.8i)23-s + (16.1 + 28.0i)25-s + (45.9 − 26.5i)29-s + (−14.6 + 25.4i)31-s + 69.0i·35-s + 18.4·37-s + (38.9 + 22.4i)41-s + (−11.5 − 19.9i)43-s + (7.32 − 4.22i)47-s + ⋯ |
| L(s) = 1 | + (−1.31 − 0.757i)5-s + (−0.651 − 1.12i)7-s + (0.0348 − 0.0201i)11-s + (0.427 − 0.740i)13-s + 0.471i·17-s − 0.427·19-s + (−0.888 − 0.513i)23-s + (0.647 + 1.12i)25-s + (1.58 − 0.913i)29-s + (−0.473 + 0.819i)31-s + 1.97i·35-s + 0.499·37-s + (0.950 + 0.548i)41-s + (−0.267 − 0.463i)43-s + (0.155 − 0.0899i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.547 + 0.836i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.547 + 0.836i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.350911 - 0.649285i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.350911 - 0.649285i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (6.55 + 3.78i)T + (12.5 + 21.6i)T^{2} \) |
| 7 | \( 1 + (4.55 + 7.89i)T + (-24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + (-0.383 + 0.221i)T + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (-5.55 + 9.62i)T + (-84.5 - 146. i)T^{2} \) |
| 17 | \( 1 - 8.01iT - 289T^{2} \) |
| 19 | \( 1 + 8.11T + 361T^{2} \) |
| 23 | \( 1 + (20.4 + 11.8i)T + (264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + (-45.9 + 26.5i)T + (420.5 - 728. i)T^{2} \) |
| 31 | \( 1 + (14.6 - 25.4i)T + (-480.5 - 832. i)T^{2} \) |
| 37 | \( 1 - 18.4T + 1.36e3T^{2} \) |
| 41 | \( 1 + (-38.9 - 22.4i)T + (840.5 + 1.45e3i)T^{2} \) |
| 43 | \( 1 + (11.5 + 19.9i)T + (-924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + (-7.32 + 4.22i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + 60.5iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (65.9 + 38.0i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (2.67 + 4.63i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-54.8 + 95.0i)T + (-2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 + 16.0iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 4.35T + 5.32e3T^{2} \) |
| 79 | \( 1 + (0.792 + 1.37i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-7.32 + 4.22i)T + (3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + 64.1iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (57.6 + 99.7i)T + (-4.70e3 + 8.14e3i)T^{2} \) |
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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
−12.93342406873095847022393732763, −12.23183158276761048839604988082, −10.98859339043037774353796527527, −10.03214952058346410208112793637, −8.489384269540464530431335121487, −7.74536939872732157704263464981, −6.39697447330836292472837099646, −4.53918976005238821994625486528, −3.57298324427366690752001843619, −0.54625460819383195273545340598,
2.81150799542191288782567888909, 4.15995692806504865553394132840, 6.02350480156748000235432463639, 7.14302323102791064878180342171, 8.362972879649692017780694826245, 9.457149691829423465927597262690, 10.88192292802850518275833364955, 11.78861517075907107398606065031, 12.48027361426847536681996231335, 13.94513067956433929254698145040
