| L(s) = 1 | + (−1.00e5 − 5.78e4i)2-s + (−4.15e7 + 1.13e7i)3-s + (4.54e9 + 7.87e9i)4-s + (−9.99e10 − 5.77e10i)5-s + (4.81e12 + 1.26e12i)6-s + (1.10e13 + 3.13e13i)7-s + (−0.0312 − 5.55e14i)8-s + (1.59e15 − 9.40e14i)9-s + (6.67e15 + 1.15e16i)10-s + (−9.36e15 + 5.40e15i)11-s + (−2.78e17 − 2.75e17i)12-s + 8.57e17·13-s + (7.04e17 − 3.78e18i)14-s + (4.80e18 + 1.26e18i)15-s + (−1.26e19 + 2.18e19i)16-s + (−1.89e19 + 1.09e19i)17-s + ⋯ |
| L(s) = 1 | + (−1.52 − 0.882i)2-s + (−0.964 + 0.263i)3-s + (1.05 + 1.83i)4-s + (−0.655 − 0.378i)5-s + (1.70 + 0.449i)6-s + (0.332 + 0.942i)7-s − 1.97i·8-s + (0.861 − 0.507i)9-s + (0.667 + 1.15i)10-s + (−0.203 + 0.117i)11-s + (−1.50 − 1.49i)12-s + 1.28·13-s + (0.323 − 1.73i)14-s + (0.731 + 0.192i)15-s + (−0.684 + 1.18i)16-s + (−0.388 + 0.224i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.784 + 0.620i)\, \overline{\Lambda}(33-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+16) \, L(s)\cr =\mathstrut & (-0.784 + 0.620i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{33}{2})\) |
\(\approx\) |
\(0.4287865417\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4287865417\) |
| \(L(17)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (4.15e7 - 1.13e7i)T \) |
| 7 | \( 1 + (-1.10e13 - 3.13e13i)T \) |
| good | 2 | \( 1 + (1.00e5 + 5.78e4i)T + (2.14e9 + 3.71e9i)T^{2} \) |
| 5 | \( 1 + (9.99e10 + 5.77e10i)T + (1.16e22 + 2.01e22i)T^{2} \) |
| 11 | \( 1 + (9.36e15 - 5.40e15i)T + (1.05e33 - 1.82e33i)T^{2} \) |
| 13 | \( 1 - 8.57e17T + 4.42e35T^{2} \) |
| 17 | \( 1 + (1.89e19 - 1.09e19i)T + (1.18e39 - 2.05e39i)T^{2} \) |
| 19 | \( 1 + (-2.54e20 + 4.40e20i)T + (-4.15e40 - 7.20e40i)T^{2} \) |
| 23 | \( 1 + (2.78e20 + 1.60e20i)T + (1.88e43 + 3.25e43i)T^{2} \) |
| 29 | \( 1 + 2.13e23iT - 6.26e46T^{2} \) |
| 31 | \( 1 + (-3.07e23 - 5.33e23i)T + (-2.64e47 + 4.58e47i)T^{2} \) |
| 37 | \( 1 + (-7.13e23 + 1.23e24i)T + (-7.61e49 - 1.31e50i)T^{2} \) |
| 41 | \( 1 + 5.55e25iT - 4.06e51T^{2} \) |
| 43 | \( 1 + 1.63e26T + 1.86e52T^{2} \) |
| 47 | \( 1 + (1.40e26 + 8.08e25i)T + (1.60e53 + 2.78e53i)T^{2} \) |
| 53 | \( 1 + (-5.45e27 + 3.14e27i)T + (7.51e54 - 1.30e55i)T^{2} \) |
| 59 | \( 1 + (-3.44e28 + 1.98e28i)T + (2.32e56 - 4.02e56i)T^{2} \) |
| 61 | \( 1 + (-2.47e28 + 4.29e28i)T + (-6.75e56 - 1.16e57i)T^{2} \) |
| 67 | \( 1 + (-5.92e28 - 1.02e29i)T + (-1.35e58 + 2.35e58i)T^{2} \) |
| 71 | \( 1 - 4.71e29iT - 1.73e59T^{2} \) |
| 73 | \( 1 + (-2.25e29 - 3.90e29i)T + (-2.11e59 + 3.66e59i)T^{2} \) |
| 79 | \( 1 + (-9.62e29 + 1.66e30i)T + (-2.64e60 - 4.58e60i)T^{2} \) |
| 83 | \( 1 + 5.34e30iT - 2.57e61T^{2} \) |
| 89 | \( 1 + (5.64e30 + 3.26e30i)T + (1.20e62 + 2.07e62i)T^{2} \) |
| 97 | \( 1 - 5.20e31T + 3.77e63T^{2} \) |
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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
−11.46151468757915769745672663216, −10.19087322744586416036478280696, −8.992947439718269078626040308914, −8.197458776477261398986380185840, −6.75125873188474405212333663600, −5.16564245812769997474234686466, −3.70391888258871367913339845542, −2.24942165715386044247999347838, −1.01948949841688274007899404352, −0.27391785600103737271908239970,
0.845231761221166168871536551727, 1.48919049765281949213648260352, 3.84986391954019683976410134440, 5.53787131416982415336289206369, 6.61844405926940339990178956654, 7.49012190765712466987379582977, 8.258481539145070122658164591815, 9.943410881979205604042855094326, 10.80909152738977485826717677724, 11.60303750150410849367635771154
