| L(s) = 1 | + (6.91e3 − 1.19e4i)2-s + (−2.39e6 − 4.14e6i)3-s + (1.72e8 + 2.99e8i)4-s + (4.10e9 − 7.10e9i)5-s − 6.61e10·6-s + (4.76e10 + 1.79e12i)7-s + (1.22e13 − 0.000732i)8-s + (−1.14e13 + 1.98e13i)9-s + (−5.67e13 − 9.83e13i)10-s + (−1.23e15 − 2.13e15i)11-s + (8.25e14 − 1.43e15i)12-s − 7.69e15·13-s + (2.18e16 + 1.18e16i)14-s − 3.92e16·15-s + (−8.21e15 + 1.42e16i)16-s + (6.53e17 + 1.13e18i)17-s + ⋯ |
| L(s) = 1 | + (0.298 − 0.517i)2-s + (−0.288 − 0.499i)3-s + (0.321 + 0.557i)4-s + (0.300 − 0.520i)5-s − 0.344·6-s + (0.0265 + 0.999i)7-s + 0.981·8-s + (−0.166 + 0.288i)9-s + (−0.179 − 0.311i)10-s + (−0.979 − 1.69i)11-s + (0.185 − 0.321i)12-s − 0.542·13-s + (0.525 + 0.284i)14-s − 0.347·15-s + (−0.0285 + 0.0493i)16-s + (0.941 + 1.63i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.575 - 0.817i)\, \overline{\Lambda}(30-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+29/2) \, L(s)\cr =\mathstrut & (-0.575 - 0.817i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(15)\) |
\(\approx\) |
\(0.3399983322\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3399983322\) |
| \(L(\frac{31}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (2.39e6 + 4.14e6i)T \) |
| 7 | \( 1 + (-4.76e10 - 1.79e12i)T \) |
| good | 2 | \( 1 + (-6.91e3 + 1.19e4i)T + (-2.68e8 - 4.64e8i)T^{2} \) |
| 5 | \( 1 + (-4.10e9 + 7.10e9i)T + (-9.31e19 - 1.61e20i)T^{2} \) |
| 11 | \( 1 + (1.23e15 + 2.13e15i)T + (-7.93e29 + 1.37e30i)T^{2} \) |
| 13 | \( 1 + 7.69e15T + 2.01e32T^{2} \) |
| 17 | \( 1 + (-6.53e17 - 1.13e18i)T + (-2.40e35 + 4.17e35i)T^{2} \) |
| 19 | \( 1 + (9.66e17 - 1.67e18i)T + (-6.06e36 - 1.05e37i)T^{2} \) |
| 23 | \( 1 + (-3.40e19 + 5.90e19i)T + (-1.54e39 - 2.67e39i)T^{2} \) |
| 29 | \( 1 + 1.17e21T + 2.56e42T^{2} \) |
| 31 | \( 1 + (2.39e21 + 4.15e21i)T + (-8.88e42 + 1.53e43i)T^{2} \) |
| 37 | \( 1 + (-2.90e22 + 5.03e22i)T + (-1.50e45 - 2.60e45i)T^{2} \) |
| 41 | \( 1 + 2.97e23T + 5.89e46T^{2} \) |
| 43 | \( 1 + 4.97e23T + 2.34e47T^{2} \) |
| 47 | \( 1 + (6.15e23 - 1.06e24i)T + (-1.54e48 - 2.68e48i)T^{2} \) |
| 53 | \( 1 + (3.55e24 + 6.16e24i)T + (-5.04e49 + 8.74e49i)T^{2} \) |
| 59 | \( 1 + (-3.16e25 - 5.48e25i)T + (-1.13e51 + 1.95e51i)T^{2} \) |
| 61 | \( 1 + (-2.72e25 + 4.72e25i)T + (-2.97e51 - 5.15e51i)T^{2} \) |
| 67 | \( 1 + (8.31e25 + 1.44e26i)T + (-4.52e52 + 7.82e52i)T^{2} \) |
| 71 | \( 1 + 9.47e26T + 4.85e53T^{2} \) |
| 73 | \( 1 + (6.07e26 + 1.05e27i)T + (-5.43e53 + 9.41e53i)T^{2} \) |
| 79 | \( 1 + (1.71e27 - 2.96e27i)T + (-5.37e54 - 9.30e54i)T^{2} \) |
| 83 | \( 1 + 3.39e27T + 4.50e55T^{2} \) |
| 89 | \( 1 + (1.10e28 - 1.90e28i)T + (-1.70e56 - 2.95e56i)T^{2} \) |
| 97 | \( 1 - 6.24e28T + 4.13e57T^{2} \) |
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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
−12.72553352668052025790548463128, −11.54865917320180093425654794645, −10.52692848985245943838276552488, −8.635018359101141998799779297924, −7.85727531204579280965026401170, −6.09739876715842099941084863484, −5.23174083099544859828743836001, −3.48821551728808091642626918952, −2.43980212697386874406356049631, −1.41403270617995861523363285800,
0.06017484550667534447102375302, 1.53764420631422385319757390557, 2.91722990584084389900088060486, 4.69359868912154774096171268210, 5.23732442203547866469806420510, 6.96348237780056965213092462332, 7.35514539919002463891698506477, 9.879020620186101918244869211432, 10.20530152475074162411465971208, 11.51337117657274547419930380355
