L(s) = 1 | + (−4.21 + 2.43i)2-s + (7.83 − 13.5i)4-s + (−6.38 − 11.0i)5-s + 37.3i·8-s + (53.7 + 31.0i)10-s + (−46.8 − 27.0i)11-s − 8.85i·13-s + (−28.1 − 48.7i)16-s + (34.4 − 59.6i)17-s + (141. − 81.9i)19-s − 200.·20-s + 263.·22-s + (81.3 − 46.9i)23-s + (−18.9 + 32.8i)25-s + (21.5 + 37.3i)26-s + ⋯ |
L(s) = 1 | + (−1.48 + 0.860i)2-s + (0.979 − 1.69i)4-s + (−0.570 − 0.988i)5-s + 1.64i·8-s + (1.70 + 0.981i)10-s + (−1.28 − 0.741i)11-s − 0.188i·13-s + (−0.439 − 0.760i)16-s + (0.491 − 0.851i)17-s + (1.71 − 0.989i)19-s − 2.23·20-s + 2.55·22-s + (0.737 − 0.425i)23-s + (−0.151 + 0.262i)25-s + (0.162 + 0.281i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.851 + 0.524i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.851 + 0.524i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.3330222936\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3330222936\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (4.21 - 2.43i)T + (4 - 6.92i)T^{2} \) |
| 5 | \( 1 + (6.38 + 11.0i)T + (-62.5 + 108. i)T^{2} \) |
| 11 | \( 1 + (46.8 + 27.0i)T + (665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + 8.85iT - 2.19e3T^{2} \) |
| 17 | \( 1 + (-34.4 + 59.6i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-141. + 81.9i)T + (3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-81.3 + 46.9i)T + (6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 - 119. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + (85.6 + 49.4i)T + (1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (47.0 + 81.5i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + 259.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 5.01T + 7.95e4T^{2} \) |
| 47 | \( 1 + (28.6 + 49.6i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-407. - 235. i)T + (7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-112. + 195. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (370. - 213. i)T + (1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (81.9 - 141. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 - 79.8iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (666. + 384. i)T + (1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (267. + 463. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + 438.T + 5.71e5T^{2} \) |
| 89 | \( 1 + (12.8 + 22.2i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + 1.38e3iT - 9.12e5T^{2} \) |
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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
−10.06615272321565919936021454467, −9.085585014722081291322184058023, −8.575283745448460511429463310466, −7.64718096834587226299817354934, −7.13029546001908125751660004781, −5.60038171916951664891977817699, −4.98686975127268318164427136364, −3.01026776436029064666294098738, −1.02080192005771003167659199757, −0.22707523569578432692393564734,
1.48584108937301688305168016180, 2.77973385123956694037844990589, 3.59241372180325014710307146787, 5.40872420682971634789250811433, 7.06612766404667101355007156635, 7.62398267004960622031569466156, 8.317377141180971038921784990377, 9.598541091283548940509993601176, 10.19034550528232183528523927463, 10.81834478573307988623030883792