| L(s) = 1 | + (67.8 + 117. i)2-s + (−3.28e3 + 5.68e3i)3-s + (5.63e4 − 9.75e4i)4-s + (1.89e5 + 3.28e5i)5-s − 8.90e5·6-s + (−1.50e7 + 2.36e6i)7-s + 3.30e7·8-s + (−2.15e7 − 3.72e7i)9-s + (−2.57e7 + 4.45e7i)10-s + (1.10e8 − 1.91e8i)11-s + (3.69e8 + 6.40e8i)12-s + 3.11e9·13-s + (−1.30e9 − 1.61e9i)14-s − 2.48e9·15-s + (−5.13e9 − 8.89e9i)16-s + (−1.91e10 + 3.32e10i)17-s + ⋯ |
| L(s) = 1 | + (0.187 + 0.324i)2-s + (−0.288 + 0.499i)3-s + (0.429 − 0.744i)4-s + (0.216 + 0.375i)5-s − 0.216·6-s + (−0.987 + 0.154i)7-s + 0.697·8-s + (−0.166 − 0.288i)9-s + (−0.0813 + 0.140i)10-s + (0.155 − 0.268i)11-s + (0.248 + 0.429i)12-s + 1.05·13-s + (−0.235 − 0.291i)14-s − 0.250·15-s + (−0.299 − 0.517i)16-s + (−0.666 + 1.15i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.908 + 0.418i)\, \overline{\Lambda}(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & (0.908 + 0.418i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(9)\) |
\(\approx\) |
\(2.007957255\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.007957255\) |
| \(L(\frac{19}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (3.28e3 - 5.68e3i)T \) |
| 7 | \( 1 + (1.50e7 - 2.36e6i)T \) |
| good | 2 | \( 1 + (-67.8 - 117. i)T + (-6.55e4 + 1.13e5i)T^{2} \) |
| 5 | \( 1 + (-1.89e5 - 3.28e5i)T + (-3.81e11 + 6.60e11i)T^{2} \) |
| 11 | \( 1 + (-1.10e8 + 1.91e8i)T + (-2.52e17 - 4.37e17i)T^{2} \) |
| 13 | \( 1 - 3.11e9T + 8.65e18T^{2} \) |
| 17 | \( 1 + (1.91e10 - 3.32e10i)T + (-4.13e20 - 7.16e20i)T^{2} \) |
| 19 | \( 1 + (5.93e10 + 1.02e11i)T + (-2.74e21 + 4.74e21i)T^{2} \) |
| 23 | \( 1 + (7.38e10 + 1.27e11i)T + (-7.05e22 + 1.22e23i)T^{2} \) |
| 29 | \( 1 - 3.50e12T + 7.25e24T^{2} \) |
| 31 | \( 1 + (-7.66e11 + 1.32e12i)T + (-1.12e25 - 1.95e25i)T^{2} \) |
| 37 | \( 1 + (4.48e12 + 7.76e12i)T + (-2.28e26 + 3.95e26i)T^{2} \) |
| 41 | \( 1 - 9.85e13T + 2.61e27T^{2} \) |
| 43 | \( 1 - 6.02e13T + 5.87e27T^{2} \) |
| 47 | \( 1 + (-7.62e13 - 1.32e14i)T + (-1.33e28 + 2.30e28i)T^{2} \) |
| 53 | \( 1 + (-3.24e14 + 5.62e14i)T + (-1.02e29 - 1.77e29i)T^{2} \) |
| 59 | \( 1 + (-4.21e14 + 7.29e14i)T + (-6.35e29 - 1.10e30i)T^{2} \) |
| 61 | \( 1 + (3.88e14 + 6.72e14i)T + (-1.12e30 + 1.94e30i)T^{2} \) |
| 67 | \( 1 + (-2.24e15 + 3.88e15i)T + (-5.52e30 - 9.56e30i)T^{2} \) |
| 71 | \( 1 + 7.43e15T + 2.96e31T^{2} \) |
| 73 | \( 1 + (5.84e15 - 1.01e16i)T + (-2.37e31 - 4.11e31i)T^{2} \) |
| 79 | \( 1 + (4.14e15 + 7.18e15i)T + (-9.09e31 + 1.57e32i)T^{2} \) |
| 83 | \( 1 - 3.40e15T + 4.21e32T^{2} \) |
| 89 | \( 1 + (1.90e16 + 3.30e16i)T + (-6.89e32 + 1.19e33i)T^{2} \) |
| 97 | \( 1 - 9.36e16T + 5.95e33T^{2} \) |
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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
−14.28213126304011282550811583347, −12.95200965108471436629673813169, −11.08507272559257923960193982377, −10.32946355045587370491528368118, −8.872759013399768062047739414441, −6.57358666986463130363606511273, −6.04767217581001987371542167670, −4.27357995696326359860452700589, −2.52378226209088146929805027843, −0.63394219958415822205109404591,
1.15617619700100131992037758098, 2.66253447726806135184274726306, 4.09346700684470761466190336393, 6.09920143604934076537849155109, 7.27281895095853910883803796901, 8.814455437287462896607056892664, 10.54270015492606587543658427675, 11.91597338579175732092993347545, 12.84506157885320516212505063571, 13.72768383312403905729266852764
