L(s) = 1 | + (0.247 − 0.429i)2-s + (1.37 − 1.05i)3-s + (0.877 + 1.51i)4-s + (−1.84 − 3.19i)5-s + (−0.109 − 0.851i)6-s + 1.86·8-s + (0.792 − 2.89i)9-s − 1.83·10-s + (0.446 − 0.772i)11-s + (2.80 + 1.17i)12-s + (−0.598 − 1.03i)13-s + (−5.90 − 2.46i)15-s + (−1.29 + 2.23i)16-s − 0.249·17-s + (−1.04 − 1.05i)18-s + 2.80·19-s + ⋯ |
L(s) = 1 | + (0.175 − 0.303i)2-s + (0.795 − 0.606i)3-s + (0.438 + 0.759i)4-s + (−0.825 − 1.43i)5-s + (−0.0447 − 0.347i)6-s + 0.658·8-s + (0.264 − 0.964i)9-s − 0.579·10-s + (0.134 − 0.233i)11-s + (0.809 + 0.337i)12-s + (−0.165 − 0.287i)13-s + (−1.52 − 0.636i)15-s + (−0.323 + 0.559i)16-s − 0.0606·17-s + (−0.246 − 0.249i)18-s + 0.644·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0816 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0816 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.41567 - 1.30449i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.41567 - 1.30449i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-1.37 + 1.05i)T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (-0.247 + 0.429i)T + (-1 - 1.73i)T^{2} \) |
| 5 | \( 1 + (1.84 + 3.19i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-0.446 + 0.772i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (0.598 + 1.03i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + 0.249T + 17T^{2} \) |
| 19 | \( 1 - 2.80T + 19T^{2} \) |
| 23 | \( 1 + (1.23 + 2.14i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.07 + 3.58i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-1.79 - 3.10i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 4.73T + 37T^{2} \) |
| 41 | \( 1 + (-2.39 - 4.14i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (4.98 - 8.64i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (5.08 - 8.81i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 9.88T + 53T^{2} \) |
| 59 | \( 1 + (-0.906 - 1.56i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-5.40 + 9.35i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (0.514 + 0.891i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 4.94T + 71T^{2} \) |
| 73 | \( 1 + 1.83T + 73T^{2} \) |
| 79 | \( 1 + (-0.899 + 1.55i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (6.16 - 10.6i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 2.40T + 89T^{2} \) |
| 97 | \( 1 + (5.52 - 9.56i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−11.39049366331349389796414199557, −9.824769735535046067043720838028, −8.791934345113698643116943724400, −8.092551829425010414274588580939, −7.63108609875043951574443726508, −6.40292874316780386603588993402, −4.75314233607271916099760193696, −3.84172644487950478505789308201, −2.74861061824448536008079892529, −1.17278024141915365176269407644,
2.21528388608458577474396997482, 3.34579123972321004116346480109, 4.39228535903513058703384262193, 5.66841003433102553596707366860, 7.02328225195842211491847694947, 7.33619995945276213862387415410, 8.538792256760748964709744771632, 9.851140935701137624373067391819, 10.30669034536320633699478064825, 11.21389010432718974992800755220