L(s) = 1 | + (1.52 + 1.29i)2-s + (0.637 + 3.94i)4-s − 7.82·5-s − 12.3i·7-s + (−4.14 + 6.83i)8-s + (−11.9 − 10.1i)10-s − 11.0i·11-s + 3.54·13-s + (16.0 − 18.8i)14-s + (−15.1 + 5.03i)16-s − 8.77·17-s − 19.2i·19-s + (−4.98 − 30.8i)20-s + (14.3 − 16.8i)22-s − 0.712i·23-s + ⋯ |
L(s) = 1 | + (0.761 + 0.648i)2-s + (0.159 + 0.987i)4-s − 1.56·5-s − 1.76i·7-s + (−0.518 + 0.854i)8-s + (−1.19 − 1.01i)10-s − 1.00i·11-s + 0.273·13-s + (1.14 − 1.34i)14-s + (−0.949 + 0.314i)16-s − 0.516·17-s − 1.01i·19-s + (−0.249 − 1.54i)20-s + (0.653 − 0.767i)22-s − 0.0309i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.159 + 0.987i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.159 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.777817 - 0.662325i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.777817 - 0.662325i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.52 - 1.29i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 7.82T + 25T^{2} \) |
| 7 | \( 1 + 12.3iT - 49T^{2} \) |
| 11 | \( 1 + 11.0iT - 121T^{2} \) |
| 13 | \( 1 - 3.54T + 169T^{2} \) |
| 17 | \( 1 + 8.77T + 289T^{2} \) |
| 19 | \( 1 + 19.2iT - 361T^{2} \) |
| 23 | \( 1 + 0.712iT - 529T^{2} \) |
| 29 | \( 1 + 18.2T + 841T^{2} \) |
| 31 | \( 1 - 11.1iT - 961T^{2} \) |
| 37 | \( 1 + 35.5T + 1.36e3T^{2} \) |
| 41 | \( 1 - 12.1T + 1.68e3T^{2} \) |
| 43 | \( 1 + 66.2iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 41.4iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 74.7T + 2.80e3T^{2} \) |
| 59 | \( 1 - 20.7iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 48.8T + 3.72e3T^{2} \) |
| 67 | \( 1 + 8.04iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 87.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 62.0T + 5.32e3T^{2} \) |
| 79 | \( 1 + 9.91iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 22.1iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 106.T + 7.92e3T^{2} \) |
| 97 | \( 1 - 131.T + 9.40e3T^{2} \) |
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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
−11.13382322723325498672414943617, −10.82758439436927498780795484020, −8.913631125823203645855889992137, −7.974728722748303044519446897660, −7.30107190856466794118375870986, −6.54306184768111840955660392371, −4.90373246276633966055448764855, −3.97040844880692093880716849637, −3.35742988412877312827266130595, −0.36748067755185164806790539332,
2.01523477691659334464871171593, 3.32312803835878458217979597440, 4.39081759992255076484578495371, 5.41159143560138946152366809546, 6.56620284228169825843141906070, 7.85525345174089889084460851411, 8.861465735442047055224048933069, 9.841876668701776324963125458033, 11.15263735196245736157874056355, 11.72510072257697963746810319088