L(s) = 1 | + (−9.02e3 − 1.56e4i)2-s + (−2.39e6 + 4.14e6i)3-s + (1.05e8 − 1.82e8i)4-s + (−6.88e9 − 1.19e10i)5-s + 8.63e10·6-s + (−1.50e12 − 9.84e11i)7-s + (−1.35e13 − 0.000244i)8-s + (−1.14e13 − 1.98e13i)9-s + (−1.24e14 + 2.15e14i)10-s + (−3.33e14 + 5.77e14i)11-s + (5.04e14 + 8.74e14i)12-s − 1.84e16·13-s + (−1.84e15 + 3.23e16i)14-s + (6.58e16 + 4i)15-s + (6.52e16 + 1.12e17i)16-s + (−2.58e17 + 4.47e17i)17-s + ⋯ |
L(s) = 1 | + (−0.389 − 0.674i)2-s + (−0.288 + 0.499i)3-s + (0.196 − 0.340i)4-s + (−0.504 − 0.873i)5-s + 0.449·6-s + (−0.836 − 0.548i)7-s − 1.08·8-s + (−0.166 − 0.288i)9-s + (−0.392 + 0.680i)10-s + (−0.264 + 0.458i)11-s + (0.113 + 0.196i)12-s − 1.29·13-s + (−0.0443 + 0.777i)14-s + 0.582·15-s + (0.226 + 0.391i)16-s + (−0.372 + 0.644i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.392 + 0.919i)\, \overline{\Lambda}(30-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+29/2) \, L(s)\cr =\mathstrut & (0.392 + 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(15)\) |
\(\approx\) |
\(0.4683427357\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4683427357\) |
\(L(\frac{31}{2})\) |
|
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 | \( 1 + (1.50e12 + 9.84e11i)T \) |
good | 2 | \( 1 + (9.02e3 + 1.56e4i)T + (-2.68e8 + 4.64e8i)T^{2} \) |
| 5 | \( 1 + (6.88e9 + 1.19e10i)T + (-9.31e19 + 1.61e20i)T^{2} \) |
| 11 | \( 1 + (3.33e14 - 5.77e14i)T + (-7.93e29 - 1.37e30i)T^{2} \) |
| 13 | \( 1 + 1.84e16T + 2.01e32T^{2} \) |
| 17 | \( 1 + (2.58e17 - 4.47e17i)T + (-2.40e35 - 4.17e35i)T^{2} \) |
| 19 | \( 1 + (-2.79e18 - 4.84e18i)T + (-6.06e36 + 1.05e37i)T^{2} \) |
| 23 | \( 1 + (1.01e19 + 1.74e19i)T + (-1.54e39 + 2.67e39i)T^{2} \) |
| 29 | \( 1 - 2.56e21T + 2.56e42T^{2} \) |
| 31 | \( 1 + (1.84e21 - 3.19e21i)T + (-8.88e42 - 1.53e43i)T^{2} \) |
| 37 | \( 1 + (3.72e22 + 6.45e22i)T + (-1.50e45 + 2.60e45i)T^{2} \) |
| 41 | \( 1 + 4.72e23T + 5.89e46T^{2} \) |
| 43 | \( 1 + 1.19e23T + 2.34e47T^{2} \) |
| 47 | \( 1 + (-4.26e23 - 7.38e23i)T + (-1.54e48 + 2.68e48i)T^{2} \) |
| 53 | \( 1 + (8.73e23 - 1.51e24i)T + (-5.04e49 - 8.74e49i)T^{2} \) |
| 59 | \( 1 + (-2.18e25 + 3.78e25i)T + (-1.13e51 - 1.95e51i)T^{2} \) |
| 61 | \( 1 + (-6.32e25 - 1.09e26i)T + (-2.97e51 + 5.15e51i)T^{2} \) |
| 67 | \( 1 + (9.73e25 - 1.68e26i)T + (-4.52e52 - 7.82e52i)T^{2} \) |
| 71 | \( 1 + 4.09e26T + 4.85e53T^{2} \) |
| 73 | \( 1 + (-4.48e25 + 7.76e25i)T + (-5.43e53 - 9.41e53i)T^{2} \) |
| 79 | \( 1 + (2.20e27 + 3.82e27i)T + (-5.37e54 + 9.30e54i)T^{2} \) |
| 83 | \( 1 + 1.03e28T + 4.50e55T^{2} \) |
| 89 | \( 1 + (1.40e28 + 2.43e28i)T + (-1.70e56 + 2.95e56i)T^{2} \) |
| 97 | \( 1 + 8.32e28T + 4.13e57T^{2} \) |
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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
−11.99730045839613585576587306049, −10.35367069674046797933130967941, −9.919466993353912374264142371835, −8.591038353668643231996759347314, −6.95635809389966502444733300568, −5.50960846674253812126053722488, −4.30411204620345339201739220778, −3.02584453319410222521883876884, −1.54961776265709561668147223159, −0.35991249753555896144029099798,
0.31447590802977448244930095959, 2.59070684542599014588000016863, 3.13105010450148996959147263494, 5.25818458642373239153432106057, 6.76338521836275193288649389477, 7.07867353195101526526125929882, 8.399970462074086148665790856785, 9.748164145752955220096635881687, 11.43379599963987192298626990073, 12.16081751614698045085794540970