| L(s) = 1 | + (627. + 362. i)2-s + (2.62e5 + 4.54e5i)4-s + (7.64e6 − 4.41e6i)5-s + (−7.20e7 + 1.24e8i)7-s + 3.79e8i·8-s + 6.39e9·10-s + (1.55e10 + 8.95e9i)11-s + (−7.25e10 − 1.25e11i)13-s + (−9.03e10 + 5.21e10i)14-s + (−1.37e11 + 2.38e11i)16-s + 2.41e12i·17-s + 1.78e12·19-s + (4.00e12 + 2.31e12i)20-s + (6.48e12 + 1.12e13i)22-s + (2.36e13 − 1.36e13i)23-s + ⋯ |
| L(s) = 1 | + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s + (0.782 − 0.451i)5-s + (−0.255 + 0.441i)7-s + 0.353i·8-s + 0.639·10-s + (0.597 + 0.345i)11-s + (−0.526 − 0.911i)13-s + (−0.312 + 0.180i)14-s + (−0.125 + 0.216i)16-s + 1.19i·17-s + 0.291·19-s + (0.391 + 0.225i)20-s + (0.244 + 0.422i)22-s + (0.569 − 0.329i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.301 - 0.953i)\, \overline{\Lambda}(21-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s+10) \, L(s)\cr =\mathstrut & (-0.301 - 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{21}{2})\) |
\(\approx\) |
\(3.243931087\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.243931087\) |
| \(L(11)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-627. - 362. i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (-7.64e6 + 4.41e6i)T + (4.76e13 - 8.25e13i)T^{2} \) |
| 7 | \( 1 + (7.20e7 - 1.24e8i)T + (-3.98e16 - 6.91e16i)T^{2} \) |
| 11 | \( 1 + (-1.55e10 - 8.95e9i)T + (3.36e20 + 5.82e20i)T^{2} \) |
| 13 | \( 1 + (7.25e10 + 1.25e11i)T + (-9.50e21 + 1.64e22i)T^{2} \) |
| 17 | \( 1 - 2.41e12iT - 4.06e24T^{2} \) |
| 19 | \( 1 - 1.78e12T + 3.75e25T^{2} \) |
| 23 | \( 1 + (-2.36e13 + 1.36e13i)T + (8.58e26 - 1.48e27i)T^{2} \) |
| 29 | \( 1 + (3.28e14 + 1.89e14i)T + (8.84e28 + 1.53e29i)T^{2} \) |
| 31 | \( 1 + (-6.77e14 - 1.17e15i)T + (-3.35e29 + 5.81e29i)T^{2} \) |
| 37 | \( 1 + 4.30e15T + 2.31e31T^{2} \) |
| 41 | \( 1 + (1.37e16 - 7.93e15i)T + (9.00e31 - 1.56e32i)T^{2} \) |
| 43 | \( 1 + (-1.44e16 + 2.50e16i)T + (-2.33e32 - 4.04e32i)T^{2} \) |
| 47 | \( 1 + (-3.84e16 - 2.21e16i)T + (1.38e33 + 2.39e33i)T^{2} \) |
| 53 | \( 1 - 1.71e16iT - 3.05e34T^{2} \) |
| 59 | \( 1 + (1.92e17 - 1.11e17i)T + (1.30e35 - 2.26e35i)T^{2} \) |
| 61 | \( 1 + (-4.11e17 + 7.12e17i)T + (-2.54e35 - 4.40e35i)T^{2} \) |
| 67 | \( 1 + (-6.59e17 - 1.14e18i)T + (-1.66e36 + 2.87e36i)T^{2} \) |
| 71 | \( 1 - 3.46e18iT - 1.05e37T^{2} \) |
| 73 | \( 1 - 4.16e18T + 1.84e37T^{2} \) |
| 79 | \( 1 + (1.16e18 - 2.02e18i)T + (-4.48e37 - 7.76e37i)T^{2} \) |
| 83 | \( 1 + (-1.17e19 - 6.78e18i)T + (1.20e38 + 2.08e38i)T^{2} \) |
| 89 | \( 1 - 7.75e18iT - 9.72e38T^{2} \) |
| 97 | \( 1 + (4.86e19 - 8.42e19i)T + (-2.71e39 - 4.70e39i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.15436965007507295389751801156, −10.56993813754745214347773154503, −9.414884359800532783670355867495, −8.325900379033057188986796578193, −6.92522247702239605125537124304, −5.82285088299432098476193746191, −5.02935835993367817792657250259, −3.64367513868321742635974376542, −2.38519421800104502033000606110, −1.22286675163458162995886753671,
0.49619738718672597783418410079, 1.78418943266161375492154866995, 2.80726329390095658780675316016, 3.97781274221442086817840440929, 5.21752078524618930926631493519, 6.40916541056405269976896937682, 7.25800336936697135105927856427, 9.207913753006489428897032326580, 9.971486966477085866519171570975, 11.19705607536163563167258322936
