| L(s) = 1 | + (1.24 + 0.671i)2-s + (−1.72 + 0.129i)3-s + (1.09 + 1.67i)4-s + (−1.37 + 0.902i)5-s + (−2.23 − 0.998i)6-s + (1.37 + 1.30i)7-s + (0.245 + 2.81i)8-s + (2.96 − 0.445i)9-s + (−2.31 + 0.202i)10-s + (0.176 + 0.351i)11-s + (−2.11 − 2.74i)12-s + (−4.46 + 1.92i)13-s + (0.843 + 2.54i)14-s + (2.25 − 1.73i)15-s + (−1.58 + 3.67i)16-s + (−2.12 − 2.52i)17-s + ⋯ |
| L(s) = 1 | + (0.880 + 0.474i)2-s + (−0.997 + 0.0745i)3-s + (0.549 + 0.835i)4-s + (−0.613 + 0.403i)5-s + (−0.913 − 0.407i)6-s + (0.521 + 0.491i)7-s + (0.0866 + 0.996i)8-s + (0.988 − 0.148i)9-s + (−0.731 + 0.0638i)10-s + (0.0531 + 0.105i)11-s + (−0.609 − 0.792i)12-s + (−1.23 + 0.534i)13-s + (0.225 + 0.680i)14-s + (0.581 − 0.448i)15-s + (−0.396 + 0.917i)16-s + (−0.514 − 0.612i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.924 - 0.380i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.924 - 0.380i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.245432 + 1.24085i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.245432 + 1.24085i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.24 - 0.671i)T \) |
| 3 | \( 1 + (1.72 - 0.129i)T \) |
| good | 5 | \( 1 + (1.37 - 0.902i)T + (1.98 - 4.59i)T^{2} \) |
| 7 | \( 1 + (-1.37 - 1.30i)T + (0.407 + 6.98i)T^{2} \) |
| 11 | \( 1 + (-0.176 - 0.351i)T + (-6.56 + 8.82i)T^{2} \) |
| 13 | \( 1 + (4.46 - 1.92i)T + (8.92 - 9.45i)T^{2} \) |
| 17 | \( 1 + (2.12 + 2.52i)T + (-2.95 + 16.7i)T^{2} \) |
| 19 | \( 1 + (1.51 + 1.27i)T + (3.29 + 18.7i)T^{2} \) |
| 23 | \( 1 + (-0.553 - 0.586i)T + (-1.33 + 22.9i)T^{2} \) |
| 29 | \( 1 + (-1.75 + 0.204i)T + (28.2 - 6.68i)T^{2} \) |
| 31 | \( 1 + (-2.11 - 8.92i)T + (-27.7 + 13.9i)T^{2} \) |
| 37 | \( 1 + (3.20 - 0.565i)T + (34.7 - 12.6i)T^{2} \) |
| 41 | \( 1 + (8.88 + 6.61i)T + (11.7 + 39.2i)T^{2} \) |
| 43 | \( 1 + (0.499 - 8.58i)T + (-42.7 - 4.99i)T^{2} \) |
| 47 | \( 1 + (-3.79 - 0.899i)T + (42.0 + 21.0i)T^{2} \) |
| 53 | \( 1 + (-1.14 - 1.98i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-2.78 + 5.55i)T + (-35.2 - 47.3i)T^{2} \) |
| 61 | \( 1 + (-6.81 + 2.03i)T + (50.9 - 33.5i)T^{2} \) |
| 67 | \( 1 + (-8.05 - 0.941i)T + (65.1 + 15.4i)T^{2} \) |
| 71 | \( 1 + (-12.9 - 4.72i)T + (54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + (2.08 - 0.758i)T + (55.9 - 46.9i)T^{2} \) |
| 79 | \( 1 + (-6.45 + 4.80i)T + (22.6 - 75.6i)T^{2} \) |
| 83 | \( 1 + (-10.1 + 7.55i)T + (23.8 - 79.5i)T^{2} \) |
| 89 | \( 1 + (-1.18 - 3.26i)T + (-68.1 + 57.2i)T^{2} \) |
| 97 | \( 1 + (-13.1 - 8.62i)T + (38.4 + 89.0i)T^{2} \) |
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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
−11.28254626285457566582597314249, −10.39881719521782442775138515069, −9.137194531699524993414989523761, −7.980274044007594527829582706037, −6.99710953949901188462851260311, −6.63141058156750722324954152028, −5.14557949370656773397140288954, −4.88408862475483227735107656143, −3.65703157197472109796350900734, −2.20806389960448476807284142753,
0.56171018418807656514353903134, 2.11993992897631507862088619201, 3.88092005449870173404230604651, 4.59228147965334354213030355260, 5.35007633665246476588394300794, 6.40378646907418050946757589947, 7.30427901882315288315633396987, 8.232599331343125543881259796359, 9.814190803296981752337371962962, 10.43176342373713208460943490753
