| L(s) = 1 | + (−7.76e4 − 4.48e4i)2-s + (−2.98e7 − 3.10e7i)3-s + (1.87e9 + 3.24e9i)4-s + (−1.87e11 − 1.08e11i)5-s + (9.25e11 + 3.74e12i)6-s + (−3.25e13 − 6.48e12i)7-s + 4.95e13i·8-s + (−7.30e13 + 1.85e15i)9-s + (9.69e15 + 1.67e16i)10-s + (6.79e16 − 3.92e16i)11-s + (4.47e16 − 1.54e17i)12-s − 9.87e16·13-s + (2.24e18 + 1.96e18i)14-s + (2.23e18 + 9.03e18i)15-s + (1.02e19 − 1.77e19i)16-s + (−1.30e18 + 7.50e17i)17-s + ⋯ |
| L(s) = 1 | + (−1.18 − 0.683i)2-s + (−0.693 − 0.720i)3-s + (0.435 + 0.754i)4-s + (−1.22 − 0.708i)5-s + (0.327 + 1.32i)6-s + (−0.980 − 0.195i)7-s + 0.176i·8-s + (−0.0394 + 0.999i)9-s + (0.969 + 1.67i)10-s + (1.47 − 0.853i)11-s + (0.242 − 0.837i)12-s − 0.148·13-s + (1.02 + 0.901i)14-s + (0.339 + 1.37i)15-s + (0.556 − 0.963i)16-s + (−0.0267 + 0.0154i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.806 - 0.591i)\, \overline{\Lambda}(33-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+16) \, L(s)\cr =\mathstrut & (0.806 - 0.591i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{33}{2})\) |
\(\approx\) |
\(0.2581399560\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2581399560\) |
| \(L(17)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (2.98e7 + 3.10e7i)T \) |
| 7 | \( 1 + (3.25e13 + 6.48e12i)T \) |
| good | 2 | \( 1 + (7.76e4 + 4.48e4i)T + (2.14e9 + 3.71e9i)T^{2} \) |
| 5 | \( 1 + (1.87e11 + 1.08e11i)T + (1.16e22 + 2.01e22i)T^{2} \) |
| 11 | \( 1 + (-6.79e16 + 3.92e16i)T + (1.05e33 - 1.82e33i)T^{2} \) |
| 13 | \( 1 + 9.87e16T + 4.42e35T^{2} \) |
| 17 | \( 1 + (1.30e18 - 7.50e17i)T + (1.18e39 - 2.05e39i)T^{2} \) |
| 19 | \( 1 + (-1.90e20 + 3.29e20i)T + (-4.15e40 - 7.20e40i)T^{2} \) |
| 23 | \( 1 + (7.02e21 + 4.05e21i)T + (1.88e43 + 3.25e43i)T^{2} \) |
| 29 | \( 1 + 9.97e22iT - 6.26e46T^{2} \) |
| 31 | \( 1 + (-4.09e23 - 7.09e23i)T + (-2.64e47 + 4.58e47i)T^{2} \) |
| 37 | \( 1 + (-1.02e25 + 1.77e25i)T + (-7.61e49 - 1.31e50i)T^{2} \) |
| 41 | \( 1 - 2.78e24iT - 4.06e51T^{2} \) |
| 43 | \( 1 - 1.29e25T + 1.86e52T^{2} \) |
| 47 | \( 1 + (3.01e26 + 1.74e26i)T + (1.60e53 + 2.78e53i)T^{2} \) |
| 53 | \( 1 + (6.15e27 - 3.55e27i)T + (7.51e54 - 1.30e55i)T^{2} \) |
| 59 | \( 1 + (1.02e28 - 5.93e27i)T + (2.32e56 - 4.02e56i)T^{2} \) |
| 61 | \( 1 + (1.19e27 - 2.06e27i)T + (-6.75e56 - 1.16e57i)T^{2} \) |
| 67 | \( 1 + (1.00e29 + 1.74e29i)T + (-1.35e58 + 2.35e58i)T^{2} \) |
| 71 | \( 1 + 1.06e29iT - 1.73e59T^{2} \) |
| 73 | \( 1 + (1.32e29 + 2.29e29i)T + (-2.11e59 + 3.66e59i)T^{2} \) |
| 79 | \( 1 + (-3.41e29 + 5.92e29i)T + (-2.64e60 - 4.58e60i)T^{2} \) |
| 83 | \( 1 + 8.37e30iT - 2.57e61T^{2} \) |
| 89 | \( 1 + (1.33e31 + 7.70e30i)T + (1.20e62 + 2.07e62i)T^{2} \) |
| 97 | \( 1 + 7.68e31T + 3.77e63T^{2} \) |
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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
−10.89441712675948707937850493681, −9.413003598568195744374012272622, −8.452947179680227760689792895138, −7.36773970920767840258769099168, −6.12613993058077644139599600655, −4.42901921841606198993848341800, −3.02687547525044582154963033695, −1.44489615139606047325729811128, −0.47426871961020797384139565780, −0.22651174885365554518270403404,
1.13858566212301987316197689821, 3.44372862342347046139796524401, 4.12376305196575902005785349541, 6.15124248895113745781544663200, 6.87007499820539305244216671991, 7.981243688208625614197735825534, 9.521634882650265689662118638536, 9.959145268938287256200845400650, 11.50474451349315191996381497929, 12.29825822233395297955497845489
