| L(s) = 1 | + i·2-s − i·3-s − 4-s + 6-s + 1.44i·7-s − i·8-s − 9-s + 2·11-s + i·12-s + 2.89i·13-s − 1.44·14-s + 16-s − 5.44i·17-s − i·18-s − 6.89·19-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.577i·3-s − 0.5·4-s + 0.408·6-s + 0.547i·7-s − 0.353i·8-s − 0.333·9-s + 0.603·11-s + 0.288i·12-s + 0.804i·13-s − 0.387·14-s + 0.250·16-s − 1.32i·17-s − 0.235i·18-s − 1.58·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4267133565\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4267133565\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + iT \) |
| 5 | \( 1 \) |
| 23 | \( 1 + iT \) |
| good | 7 | \( 1 - 1.44iT - 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 - 2.89iT - 13T^{2} \) |
| 17 | \( 1 + 5.44iT - 17T^{2} \) |
| 19 | \( 1 + 6.89T + 19T^{2} \) |
| 29 | \( 1 + 5T + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 - 8.34iT - 37T^{2} \) |
| 41 | \( 1 + 4.89T + 41T^{2} \) |
| 43 | \( 1 + 10.8iT - 43T^{2} \) |
| 47 | \( 1 + 3.89iT - 47T^{2} \) |
| 53 | \( 1 + 0.898iT - 53T^{2} \) |
| 59 | \( 1 + 10T + 59T^{2} \) |
| 61 | \( 1 + 4.89T + 61T^{2} \) |
| 67 | \( 1 + 2iT - 67T^{2} \) |
| 71 | \( 1 - 10.7T + 71T^{2} \) |
| 73 | \( 1 + 2.10iT - 73T^{2} \) |
| 79 | \( 1 + 10T + 79T^{2} \) |
| 83 | \( 1 - 2.55iT - 83T^{2} \) |
| 89 | \( 1 - 10.3T + 89T^{2} \) |
| 97 | \( 1 + 12.6iT - 97T^{2} \) |
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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
−8.398804470648847736577494169222, −7.48043210064073063700890741386, −6.74630733547011031578990073090, −6.37851870507351190960917108224, −5.43587399989553101673709044942, −4.66598590075881273390266737024, −3.78408643474577313001808790794, −2.59437462966446178650980618589, −1.64867250951098404296743373905, −0.12887138816868889017460606192,
1.31504803603912228295738623817, 2.38258528050711413442313461305, 3.52220371933150108422045865483, 4.02692150389624783222978551345, 4.74322255338198094055449611221, 5.81963619628727021680337569750, 6.38846904223840871050809471889, 7.55722270992837872677010268871, 8.262954188722980875997673108152, 8.941332732924538487143364192437
