| L(s) = 1 | + (1.59 − 4.90i)3-s + (−1.80 + 1.31i)5-s + (−9.66 + 3.13i)7-s + (−14.2 − 10.3i)9-s + (−1.51 − 10.8i)11-s + (−1.96 + 2.70i)13-s + (3.56 + 10.9i)15-s + (−11.8 − 16.3i)17-s + (−5.26 − 1.70i)19-s + 52.3i·21-s + 34.6·23-s + (1.54 − 4.75i)25-s + (−35.9 + 26.1i)27-s + (−42.6 + 13.8i)29-s + (35.0 + 25.4i)31-s + ⋯ |
| L(s) = 1 | + (0.531 − 1.63i)3-s + (−0.361 + 0.262i)5-s + (−1.38 + 0.448i)7-s + (−1.58 − 1.15i)9-s + (−0.137 − 0.990i)11-s + (−0.151 + 0.208i)13-s + (0.237 + 0.731i)15-s + (−0.697 − 0.960i)17-s + (−0.276 − 0.0899i)19-s + 2.49i·21-s + 1.50·23-s + (0.0618 − 0.190i)25-s + (−1.33 + 0.967i)27-s + (−1.47 + 0.478i)29-s + (1.13 + 0.821i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0682i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.997 + 0.0682i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0307434 - 0.900053i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0307434 - 0.900053i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (1.80 - 1.31i)T \) |
| 11 | \( 1 + (1.51 + 10.8i)T \) |
| good | 3 | \( 1 + (-1.59 + 4.90i)T + (-7.28 - 5.29i)T^{2} \) |
| 7 | \( 1 + (9.66 - 3.13i)T + (39.6 - 28.8i)T^{2} \) |
| 13 | \( 1 + (1.96 - 2.70i)T + (-52.2 - 160. i)T^{2} \) |
| 17 | \( 1 + (11.8 + 16.3i)T + (-89.3 + 274. i)T^{2} \) |
| 19 | \( 1 + (5.26 + 1.70i)T + (292. + 212. i)T^{2} \) |
| 23 | \( 1 - 34.6T + 529T^{2} \) |
| 29 | \( 1 + (42.6 - 13.8i)T + (680. - 494. i)T^{2} \) |
| 31 | \( 1 + (-35.0 - 25.4i)T + (296. + 913. i)T^{2} \) |
| 37 | \( 1 + (18.3 + 56.4i)T + (-1.10e3 + 804. i)T^{2} \) |
| 41 | \( 1 + (-31.4 - 10.2i)T + (1.35e3 + 988. i)T^{2} \) |
| 43 | \( 1 + 65.3iT - 1.84e3T^{2} \) |
| 47 | \( 1 + (-21.2 + 65.4i)T + (-1.78e3 - 1.29e3i)T^{2} \) |
| 53 | \( 1 + (-2.60 - 1.89i)T + (868. + 2.67e3i)T^{2} \) |
| 59 | \( 1 + (-20.4 - 62.7i)T + (-2.81e3 + 2.04e3i)T^{2} \) |
| 61 | \( 1 + (27.8 + 38.3i)T + (-1.14e3 + 3.53e3i)T^{2} \) |
| 67 | \( 1 + 31.6T + 4.48e3T^{2} \) |
| 71 | \( 1 + (3.04 - 2.21i)T + (1.55e3 - 4.79e3i)T^{2} \) |
| 73 | \( 1 + (-21.3 + 6.92i)T + (4.31e3 - 3.13e3i)T^{2} \) |
| 79 | \( 1 + (45.9 - 63.2i)T + (-1.92e3 - 5.93e3i)T^{2} \) |
| 83 | \( 1 + (-13.0 - 17.9i)T + (-2.12e3 + 6.55e3i)T^{2} \) |
| 89 | \( 1 - 68.2T + 7.92e3T^{2} \) |
| 97 | \( 1 + (49.7 + 36.1i)T + (2.90e3 + 8.94e3i)T^{2} \) |
| 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
−11.87607019457032770446411850609, −10.84876907416488868999299067205, −9.177000852099654230605705394741, −8.658788354823533217882761788259, −7.23201988598608559329256400971, −6.81049906599984980476043385554, −5.67826082952489845059350479843, −3.36172093025384440815976477627, −2.47668865785485053276894684458, −0.44189440520618971610213197338,
2.89026941993046505290419078596, 3.97868860692315592050671687362, 4.76226564840442262394003692172, 6.31405260823126374577265091212, 7.71601481030485966094342774214, 8.964170056922250568273295013790, 9.656074836148478003387740469409, 10.32865402230244894089653105657, 11.24986825301565995443924783381, 12.73756668243342909435661654800
