| L(s) = 1 | + i·2-s − 4-s − 5-s − i·8-s − i·10-s − 4.93i·11-s − 2.49i·13-s + 16-s + 0.433·17-s + 7.93i·19-s + 20-s + 4.93·22-s − 6.02i·23-s + 25-s + 2.49·26-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.5·4-s − 0.447·5-s − 0.353i·8-s − 0.316i·10-s − 1.48i·11-s − 0.692i·13-s + 0.250·16-s + 0.105·17-s + 1.82i·19-s + 0.223·20-s + 1.05·22-s − 1.25i·23-s + 0.200·25-s + 0.489·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.698 + 0.716i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.698 + 0.716i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2492556122\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2492556122\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 4.93iT - 11T^{2} \) |
| 13 | \( 1 + 2.49iT - 13T^{2} \) |
| 17 | \( 1 - 0.433T + 17T^{2} \) |
| 19 | \( 1 - 7.93iT - 19T^{2} \) |
| 23 | \( 1 + 6.02iT - 23T^{2} \) |
| 29 | \( 1 - 4.71iT - 29T^{2} \) |
| 31 | \( 1 + 5.75iT - 31T^{2} \) |
| 37 | \( 1 + 7.12T + 37T^{2} \) |
| 41 | \( 1 - 8.73T + 41T^{2} \) |
| 43 | \( 1 + 8.32T + 43T^{2} \) |
| 47 | \( 1 - 5.40T + 47T^{2} \) |
| 53 | \( 1 - 7.27iT - 53T^{2} \) |
| 59 | \( 1 + 3.05T + 59T^{2} \) |
| 61 | \( 1 + 3.10iT - 61T^{2} \) |
| 67 | \( 1 + 14.2T + 67T^{2} \) |
| 71 | \( 1 - 2.47iT - 71T^{2} \) |
| 73 | \( 1 - 5.71iT - 73T^{2} \) |
| 79 | \( 1 + 4.32T + 79T^{2} \) |
| 83 | \( 1 - 11.1T + 83T^{2} \) |
| 89 | \( 1 + 14.5T + 89T^{2} \) |
| 97 | \( 1 + 3.65iT - 97T^{2} \) |
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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
−8.092897266896079146653771837549, −7.51045108717422568646677086639, −6.54652039523515202169173121152, −5.86971342725869198550531701952, −5.39816885011651301427087244021, −4.28972159942702174676370017634, −3.60098087789589198632750051088, −2.81355204702839408077514710918, −1.24430871422931290334615307065, −0.07502500084070810489418383743,
1.40511077860675574892046896990, 2.28258389093158780167879726937, 3.19039725440386355208908126755, 4.15815672343710075449457315538, 4.70579990943867768571110003050, 5.41309369279491403944716451421, 6.68298732971822599834849784911, 7.17763447194287369361198380847, 7.87538929947483305738685888496, 8.899365702867642784560177476597
