| L(s) = 1 | + (0.247 + 1.98i)2-s + (−3.87 + 0.980i)4-s − 2.20·5-s − 8.35i·7-s + (−2.90 − 7.45i)8-s + (−0.544 − 4.36i)10-s + 5.25i·11-s + 14.7·13-s + (16.5 − 2.06i)14-s + (14.0 − 7.60i)16-s + 28.2·17-s − 19.1i·19-s + (8.53 − 2.15i)20-s + (−10.4 + 1.29i)22-s + 3.65i·23-s + ⋯ |
| L(s) = 1 | + (0.123 + 0.992i)2-s + (−0.969 + 0.245i)4-s − 0.440·5-s − 1.19i·7-s + (−0.363 − 0.931i)8-s + (−0.0544 − 0.436i)10-s + 0.477i·11-s + 1.13·13-s + (1.18 − 0.147i)14-s + (0.879 − 0.475i)16-s + 1.66·17-s − 1.00i·19-s + (0.426 − 0.107i)20-s + (−0.473 + 0.0589i)22-s + 0.158i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.969 - 0.245i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.969 - 0.245i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.41932 + 0.176712i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.41932 + 0.176712i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.247 - 1.98i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 2.20T + 25T^{2} \) |
| 7 | \( 1 + 8.35iT - 49T^{2} \) |
| 11 | \( 1 - 5.25iT - 121T^{2} \) |
| 13 | \( 1 - 14.7T + 169T^{2} \) |
| 17 | \( 1 - 28.2T + 289T^{2} \) |
| 19 | \( 1 + 19.1iT - 361T^{2} \) |
| 23 | \( 1 - 3.65iT - 529T^{2} \) |
| 29 | \( 1 - 24.6T + 841T^{2} \) |
| 31 | \( 1 + 38.0iT - 961T^{2} \) |
| 37 | \( 1 + 4.21T + 1.36e3T^{2} \) |
| 41 | \( 1 - 19.8T + 1.68e3T^{2} \) |
| 43 | \( 1 + 23.3iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 29.8iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 32.1T + 2.80e3T^{2} \) |
| 59 | \( 1 + 9.19iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 81.6T + 3.72e3T^{2} \) |
| 67 | \( 1 - 7.92iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 62.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 33.3T + 5.32e3T^{2} \) |
| 79 | \( 1 + 62.0iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 118. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 107.T + 7.92e3T^{2} \) |
| 97 | \( 1 + 3.57T + 9.40e3T^{2} \) |
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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
−11.46480900570988910592993146417, −10.32386596135164991264647785774, −9.498980384446105711545226354399, −8.245467106212198710592459954433, −7.57384882855032184253172768286, −6.72962241552414220759598754040, −5.57883259710538859639896305471, −4.32176700822171224470145320251, −3.52851487323532191580642910567, −0.793034492184583340968955691538,
1.32194231028093260926574485004, 2.94369608194720709518345101877, 3.85271010372455451105295257947, 5.35304663822340796149693400766, 6.07732814455367148737045748361, 8.026833168409375842735688680648, 8.569415693073807975663070211227, 9.630155124717580134444789407651, 10.54438270529297445551941207005, 11.53346972779915393315262698467
