| L(s) = 1 | + (−8.99e4 + 5.19e4i)2-s + (−1.34e6 + 4.30e7i)3-s + (3.25e9 − 5.63e9i)4-s + (2.01e11 − 1.16e11i)5-s + (−2.11e12 − 3.94e12i)6-s + (2.96e13 + 1.50e13i)7-s + (−0.0625 + 2.29e14i)8-s + (−1.84e15 − 1.16e14i)9-s + (−1.20e16 + 2.09e16i)10-s + (2.24e16 + 1.29e16i)11-s + (2.37e17 + 1.47e17i)12-s + 7.52e17·13-s + (−3.44e18 + 1.85e17i)14-s + (4.72e18 + 8.81e18i)15-s + (2.04e18 + 3.54e18i)16-s + (6.31e19 + 3.64e19i)17-s + ⋯ |
| L(s) = 1 | + (−1.37 + 0.792i)2-s + (−0.0313 + 0.999i)3-s + (0.756 − 1.31i)4-s + (1.31 − 0.761i)5-s + (−0.749 − 1.39i)6-s + (0.891 + 0.452i)7-s + 0.814i·8-s + (−0.998 − 0.0626i)9-s + (−1.20 + 2.09i)10-s + (0.488 + 0.281i)11-s + (1.28 + 0.797i)12-s + 1.13·13-s + (−1.58 + 0.0851i)14-s + (0.719 + 1.34i)15-s + (0.111 + 0.192i)16-s + (1.29 + 0.749i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.844 - 0.535i)\, \overline{\Lambda}(33-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+16) \, L(s)\cr =\mathstrut & (-0.844 - 0.535i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{33}{2})\) |
\(\approx\) |
\(1.798456350\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.798456350\) |
| \(L(17)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.34e6 - 4.30e7i)T \) |
| 7 | \( 1 + (-2.96e13 - 1.50e13i)T \) |
| good | 2 | \( 1 + (8.99e4 - 5.19e4i)T + (2.14e9 - 3.71e9i)T^{2} \) |
| 5 | \( 1 + (-2.01e11 + 1.16e11i)T + (1.16e22 - 2.01e22i)T^{2} \) |
| 11 | \( 1 + (-2.24e16 - 1.29e16i)T + (1.05e33 + 1.82e33i)T^{2} \) |
| 13 | \( 1 - 7.52e17T + 4.42e35T^{2} \) |
| 17 | \( 1 + (-6.31e19 - 3.64e19i)T + (1.18e39 + 2.05e39i)T^{2} \) |
| 19 | \( 1 + (-1.80e20 - 3.13e20i)T + (-4.15e40 + 7.20e40i)T^{2} \) |
| 23 | \( 1 + (9.36e21 - 5.40e21i)T + (1.88e43 - 3.25e43i)T^{2} \) |
| 29 | \( 1 + 3.72e22iT - 6.26e46T^{2} \) |
| 31 | \( 1 + (-1.08e23 + 1.87e23i)T + (-2.64e47 - 4.58e47i)T^{2} \) |
| 37 | \( 1 + (1.87e24 + 3.24e24i)T + (-7.61e49 + 1.31e50i)T^{2} \) |
| 41 | \( 1 - 4.32e25iT - 4.06e51T^{2} \) |
| 43 | \( 1 + 1.56e26T + 1.86e52T^{2} \) |
| 47 | \( 1 + (-7.84e26 + 4.52e26i)T + (1.60e53 - 2.78e53i)T^{2} \) |
| 53 | \( 1 + (-2.68e27 - 1.55e27i)T + (7.51e54 + 1.30e55i)T^{2} \) |
| 59 | \( 1 + (1.68e28 + 9.74e27i)T + (2.32e56 + 4.02e56i)T^{2} \) |
| 61 | \( 1 + (-2.72e28 - 4.72e28i)T + (-6.75e56 + 1.16e57i)T^{2} \) |
| 67 | \( 1 + (6.24e28 - 1.08e29i)T + (-1.35e58 - 2.35e58i)T^{2} \) |
| 71 | \( 1 + 3.91e29iT - 1.73e59T^{2} \) |
| 73 | \( 1 + (4.27e29 - 7.40e29i)T + (-2.11e59 - 3.66e59i)T^{2} \) |
| 79 | \( 1 + (-3.45e29 - 5.99e29i)T + (-2.64e60 + 4.58e60i)T^{2} \) |
| 83 | \( 1 + 3.53e30iT - 2.57e61T^{2} \) |
| 89 | \( 1 + (-1.33e30 + 7.68e29i)T + (1.20e62 - 2.07e62i)T^{2} \) |
| 97 | \( 1 + 8.26e31T + 3.77e63T^{2} \) |
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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
−11.91598059098820386663429597675, −10.32010073950676043947548928627, −9.704460010948845154457086406508, −8.742584010561182825787705985134, −7.998469121152134492346426888320, −5.85904293532244417279432446924, −5.65288762320470088012125738931, −3.89292053880972678418155483960, −1.75581875960202724173098944031, −1.15853371154334468122231063665,
0.64195769129595718908330867365, 1.34709793408000911715314323332, 2.13878169415642277071650458510, 3.14324911298073741688794101311, 5.58256661079160131162035777474, 6.79173300323306254926418348534, 7.900494510289291559645923023311, 8.958276280559040906223025757557, 10.17350356965665209200441843772, 11.07866008199388690769714270518
