L(s) = 1 | + (−2.59 + 1.50i)3-s + (−2.59 − 2.59i)5-s + 7.30i·7-s + (4.47 − 7.81i)9-s + (−11.3 − 11.3i)11-s + (0.746 + 0.746i)13-s + (10.6 + 2.83i)15-s − 6.67i·17-s + (22.1 + 22.1i)19-s + (−10.9 − 18.9i)21-s + 21.4·23-s − 11.4i·25-s + (0.153 + 26.9i)27-s + (1.54 − 1.54i)29-s + 14.6·31-s + ⋯ |
L(s) = 1 | + (−0.865 + 0.501i)3-s + (−0.519 − 0.519i)5-s + 1.04i·7-s + (0.496 − 0.867i)9-s + (−1.02 − 1.02i)11-s + (0.0574 + 0.0574i)13-s + (0.710 + 0.188i)15-s − 0.392i·17-s + (1.16 + 1.16i)19-s + (−0.523 − 0.902i)21-s + 0.932·23-s − 0.459i·25-s + (0.00567 + 0.999i)27-s + (0.0531 − 0.0531i)29-s + 0.471·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 + 0.249i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.968 + 0.249i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.983819 - 0.124583i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.983819 - 0.124583i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (2.59 - 1.50i)T \) |
good | 5 | \( 1 + (2.59 + 2.59i)T + 25iT^{2} \) |
| 7 | \( 1 - 7.30iT - 49T^{2} \) |
| 11 | \( 1 + (11.3 + 11.3i)T + 121iT^{2} \) |
| 13 | \( 1 + (-0.746 - 0.746i)T + 169iT^{2} \) |
| 17 | \( 1 + 6.67iT - 289T^{2} \) |
| 19 | \( 1 + (-22.1 - 22.1i)T + 361iT^{2} \) |
| 23 | \( 1 - 21.4T + 529T^{2} \) |
| 29 | \( 1 + (-1.54 + 1.54i)T - 841iT^{2} \) |
| 31 | \( 1 - 14.6T + 961T^{2} \) |
| 37 | \( 1 + (-50.1 + 50.1i)T - 1.36e3iT^{2} \) |
| 41 | \( 1 - 15.0T + 1.68e3T^{2} \) |
| 43 | \( 1 + (-26.3 + 26.3i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + 36.6iT - 2.20e3T^{2} \) |
| 53 | \( 1 + (-50.9 - 50.9i)T + 2.80e3iT^{2} \) |
| 59 | \( 1 + (-12.1 - 12.1i)T + 3.48e3iT^{2} \) |
| 61 | \( 1 + (-27.5 - 27.5i)T + 3.72e3iT^{2} \) |
| 67 | \( 1 + (4.84 + 4.84i)T + 4.48e3iT^{2} \) |
| 71 | \( 1 + 74.9T + 5.04e3T^{2} \) |
| 73 | \( 1 - 3.47iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 103.T + 6.24e3T^{2} \) |
| 83 | \( 1 + (-31.7 + 31.7i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 - 78.2T + 7.92e3T^{2} \) |
| 97 | \( 1 + 61.5T + 9.40e3T^{2} \) |
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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
−11.19870914641707395020544864649, −10.28805471239380926983371805286, −9.248731090030229847426797076683, −8.434378095294910085332649883042, −7.37192219874334287567317230817, −5.79219550632887294607315086370, −5.48746436459730761422867132022, −4.23822617953702751274618553098, −2.89579713722635360191509203109, −0.67970958113311047006855299424,
0.962872146062409076762713723734, 2.80215535493993949452762275944, 4.38235530154973881274699158279, 5.23862396053260267204761313388, 6.66857042814489039491298426011, 7.33726937712598065325307976852, 7.892395884977301634638700440735, 9.619186081727078870777096450011, 10.50245109375308305314913268441, 11.13589338384513236134015844418