| L(s) = 1 | + (−8.11e4 − 4.68e4i)2-s + (2.92e7 + 3.15e7i)3-s + (2.24e9 + 3.88e9i)4-s + (1.46e10 + 8.48e9i)5-s + (−8.93e11 − 3.93e12i)6-s + (5.53e12 − 3.27e13i)7-s + (0.0312 − 1.77e13i)8-s + (−1.42e14 + 1.84e15i)9-s + (−7.95e14 − 1.37e15i)10-s + (3.46e16 − 2.00e16i)11-s + (−5.70e16 + 1.84e17i)12-s + 7.68e17·13-s + (−1.98e18 + 2.39e18i)14-s + (1.61e17 + 7.12e17i)15-s + (8.79e18 − 1.52e19i)16-s + (−5.61e19 + 3.23e19i)17-s + ⋯ |
| L(s) = 1 | + (−1.23 − 0.714i)2-s + (0.679 + 0.733i)3-s + (0.522 + 0.904i)4-s + (0.0963 + 0.0556i)5-s + (−0.316 − 1.39i)6-s + (0.166 − 0.986i)7-s − 0.0631i·8-s + (−0.0767 + 0.997i)9-s + (−0.0795 − 0.137i)10-s + (0.754 − 0.435i)11-s + (−0.308 + 0.997i)12-s + 1.15·13-s + (−0.911 + 1.10i)14-s + (0.0246 + 0.108i)15-s + (0.476 − 0.826i)16-s + (−1.15 + 0.665i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.599 - 0.800i)\, \overline{\Lambda}(33-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+16) \, L(s)\cr =\mathstrut & (-0.599 - 0.800i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{33}{2})\) |
\(\approx\) |
\(0.4314976833\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4314976833\) |
| \(L(17)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-2.92e7 - 3.15e7i)T \) |
| 7 | \( 1 + (-5.53e12 + 3.27e13i)T \) |
| good | 2 | \( 1 + (8.11e4 + 4.68e4i)T + (2.14e9 + 3.71e9i)T^{2} \) |
| 5 | \( 1 + (-1.46e10 - 8.48e9i)T + (1.16e22 + 2.01e22i)T^{2} \) |
| 11 | \( 1 + (-3.46e16 + 2.00e16i)T + (1.05e33 - 1.82e33i)T^{2} \) |
| 13 | \( 1 - 7.68e17T + 4.42e35T^{2} \) |
| 17 | \( 1 + (5.61e19 - 3.23e19i)T + (1.18e39 - 2.05e39i)T^{2} \) |
| 19 | \( 1 + (9.39e19 - 1.62e20i)T + (-4.15e40 - 7.20e40i)T^{2} \) |
| 23 | \( 1 + (5.77e21 + 3.33e21i)T + (1.88e43 + 3.25e43i)T^{2} \) |
| 29 | \( 1 + 2.09e23iT - 6.26e46T^{2} \) |
| 31 | \( 1 + (-4.76e23 - 8.24e23i)T + (-2.64e47 + 4.58e47i)T^{2} \) |
| 37 | \( 1 + (1.05e25 - 1.82e25i)T + (-7.61e49 - 1.31e50i)T^{2} \) |
| 41 | \( 1 - 5.73e25iT - 4.06e51T^{2} \) |
| 43 | \( 1 - 2.04e26T + 1.86e52T^{2} \) |
| 47 | \( 1 + (9.38e26 + 5.41e26i)T + (1.60e53 + 2.78e53i)T^{2} \) |
| 53 | \( 1 + (-4.08e27 + 2.35e27i)T + (7.51e54 - 1.30e55i)T^{2} \) |
| 59 | \( 1 + (2.39e28 - 1.38e28i)T + (2.32e56 - 4.02e56i)T^{2} \) |
| 61 | \( 1 + (-1.45e28 + 2.51e28i)T + (-6.75e56 - 1.16e57i)T^{2} \) |
| 67 | \( 1 + (-9.49e28 - 1.64e29i)T + (-1.35e58 + 2.35e58i)T^{2} \) |
| 71 | \( 1 - 3.98e28iT - 1.73e59T^{2} \) |
| 73 | \( 1 + (-1.07e29 - 1.86e29i)T + (-2.11e59 + 3.66e59i)T^{2} \) |
| 79 | \( 1 + (-8.54e27 + 1.47e28i)T + (-2.64e60 - 4.58e60i)T^{2} \) |
| 83 | \( 1 + 9.34e30iT - 2.57e61T^{2} \) |
| 89 | \( 1 + (-8.59e30 - 4.96e30i)T + (1.20e62 + 2.07e62i)T^{2} \) |
| 97 | \( 1 + 3.16e31T + 3.77e63T^{2} \) |
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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
−11.56653863798416176128564758604, −10.57679173896430158168677333809, −9.975344263815356829033115300200, −8.604536521481427334773589071253, −8.187749162054895793727356044597, −6.38914750936075622003586727604, −4.38412305376759733567707342950, −3.47920295273091520676476780764, −2.08722036990986167038643315026, −1.17173707897077193785767324667,
0.12440182926338999049833533138, 1.35987183490324747074434216588, 2.22954323103022507311268998744, 3.84891698559000561807536198298, 5.92856703623050456622983722042, 6.82600042489170397381383527992, 7.901559800050851082001191884372, 9.006702299481172382574052299949, 9.296301362561574201159411318483, 11.24077762923709841711162642718
