| L(s) = 1 | + (−9.26e4 − 5.34e4i)2-s + (−4.06e7 − 1.41e7i)3-s + (3.57e9 + 6.18e9i)4-s + (2.33e11 + 1.34e11i)5-s + (3.00e12 + 3.48e12i)6-s + (−1.79e13 + 2.79e13i)7-s − 3.05e14i·8-s + (1.45e15 + 1.15e15i)9-s + (−1.44e16 − 2.49e16i)10-s + (−1.12e16 + 6.49e15i)11-s + (−5.73e16 − 3.02e17i)12-s − 8.58e17·13-s + (3.15e18 − 1.62e18i)14-s + (−7.56e18 − 8.78e18i)15-s + (−9.69e17 + 1.67e18i)16-s + (3.54e18 − 2.04e18i)17-s + ⋯ |
| L(s) = 1 | + (−1.41 − 0.816i)2-s + (−0.944 − 0.329i)3-s + (0.832 + 1.44i)4-s + (1.52 + 0.882i)5-s + (1.06 + 1.23i)6-s + (−0.540 + 0.841i)7-s − 1.08i·8-s + (0.782 + 0.622i)9-s + (−1.44 − 2.49i)10-s + (−0.244 + 0.141i)11-s + (−0.310 − 1.63i)12-s − 1.28·13-s + (1.45 − 0.747i)14-s + (−1.15 − 1.33i)15-s + (−0.0525 + 0.0910i)16-s + (0.0727 − 0.0420i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.664 - 0.747i)\, \overline{\Lambda}(33-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+16) \, L(s)\cr =\mathstrut & (-0.664 - 0.747i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{33}{2})\) |
\(\approx\) |
\(0.5807717370\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5807717370\) |
| \(L(17)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (4.06e7 + 1.41e7i)T \) |
| 7 | \( 1 + (1.79e13 - 2.79e13i)T \) |
| good | 2 | \( 1 + (9.26e4 + 5.34e4i)T + (2.14e9 + 3.71e9i)T^{2} \) |
| 5 | \( 1 + (-2.33e11 - 1.34e11i)T + (1.16e22 + 2.01e22i)T^{2} \) |
| 11 | \( 1 + (1.12e16 - 6.49e15i)T + (1.05e33 - 1.82e33i)T^{2} \) |
| 13 | \( 1 + 8.58e17T + 4.42e35T^{2} \) |
| 17 | \( 1 + (-3.54e18 + 2.04e18i)T + (1.18e39 - 2.05e39i)T^{2} \) |
| 19 | \( 1 + (6.58e19 - 1.14e20i)T + (-4.15e40 - 7.20e40i)T^{2} \) |
| 23 | \( 1 + (2.63e21 + 1.51e21i)T + (1.88e43 + 3.25e43i)T^{2} \) |
| 29 | \( 1 - 2.10e23iT - 6.26e46T^{2} \) |
| 31 | \( 1 + (-5.97e23 - 1.03e24i)T + (-2.64e47 + 4.58e47i)T^{2} \) |
| 37 | \( 1 + (-5.36e24 + 9.29e24i)T + (-7.61e49 - 1.31e50i)T^{2} \) |
| 41 | \( 1 - 6.82e25iT - 4.06e51T^{2} \) |
| 43 | \( 1 - 2.33e26T + 1.86e52T^{2} \) |
| 47 | \( 1 + (-1.75e26 - 1.01e26i)T + (1.60e53 + 2.78e53i)T^{2} \) |
| 53 | \( 1 + (-2.80e27 + 1.61e27i)T + (7.51e54 - 1.30e55i)T^{2} \) |
| 59 | \( 1 + (1.32e28 - 7.67e27i)T + (2.32e56 - 4.02e56i)T^{2} \) |
| 61 | \( 1 + (3.21e28 - 5.56e28i)T + (-6.75e56 - 1.16e57i)T^{2} \) |
| 67 | \( 1 + (-9.46e28 - 1.63e29i)T + (-1.35e58 + 2.35e58i)T^{2} \) |
| 71 | \( 1 - 3.45e29iT - 1.73e59T^{2} \) |
| 73 | \( 1 + (-1.26e29 - 2.18e29i)T + (-2.11e59 + 3.66e59i)T^{2} \) |
| 79 | \( 1 + (-8.05e29 + 1.39e30i)T + (-2.64e60 - 4.58e60i)T^{2} \) |
| 83 | \( 1 + 2.12e30iT - 2.57e61T^{2} \) |
| 89 | \( 1 + (-1.57e31 - 9.06e30i)T + (1.20e62 + 2.07e62i)T^{2} \) |
| 97 | \( 1 - 1.13e32T + 3.77e63T^{2} \) |
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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
−11.94139546550201955736445427803, −10.55736204113928506914932011436, −10.07827208331414510987639108422, −9.131144498592466060186811079571, −7.40986433492603370743710721140, −6.34355454549665646656821395398, −5.29573266312666161997639643962, −2.70832132612707425304648118722, −2.22041492575924480397134849927, −1.13420242615349664404396894530,
0.31015054086137266377442753463, 0.78062146472783535782969431435, 2.07316221520820846541678127574, 4.51887971249585092282441658966, 5.73616808214955019796687846097, 6.46937622384535187517336532381, 7.68539978784456579271240703073, 9.383586198636053745204296644813, 9.758593564005223695995416514080, 10.63286763276529578917763514145
