| L(s) = 1 | + i·2-s + 3-s − 4-s + i·5-s + i·6-s + 2.82i·7-s − i·8-s + 9-s − 10-s − 5.65i·11-s − 12-s − 2.82·14-s + i·15-s + 16-s − 0.828·17-s + i·18-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s + 0.577·3-s − 0.5·4-s + 0.447i·5-s + 0.408i·6-s + 1.06i·7-s − 0.353i·8-s + 0.333·9-s − 0.316·10-s − 1.70i·11-s − 0.288·12-s − 0.755·14-s + 0.258i·15-s + 0.250·16-s − 0.200·17-s + 0.235i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.832 - 0.554i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.832 - 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.861668220\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.861668220\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 5 | \( 1 - iT \) |
| 13 | \( 1 \) |
| good | 7 | \( 1 - 2.82iT - 7T^{2} \) |
| 11 | \( 1 + 5.65iT - 11T^{2} \) |
| 17 | \( 1 + 0.828T + 17T^{2} \) |
| 19 | \( 1 - 2.82iT - 19T^{2} \) |
| 23 | \( 1 - 8.48T + 23T^{2} \) |
| 29 | \( 1 + 8.82T + 29T^{2} \) |
| 31 | \( 1 - 4iT - 31T^{2} \) |
| 37 | \( 1 - 11.6iT - 37T^{2} \) |
| 41 | \( 1 + 7.65iT - 41T^{2} \) |
| 43 | \( 1 + 9.65T + 43T^{2} \) |
| 47 | \( 1 - 8iT - 47T^{2} \) |
| 53 | \( 1 - 13.3T + 53T^{2} \) |
| 59 | \( 1 - 2.34iT - 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 - 5.65iT - 67T^{2} \) |
| 71 | \( 1 + 5.65iT - 71T^{2} \) |
| 73 | \( 1 - 14.4iT - 73T^{2} \) |
| 79 | \( 1 - 2.34T + 79T^{2} \) |
| 83 | \( 1 - 6.34iT - 83T^{2} \) |
| 89 | \( 1 - 15.6iT - 89T^{2} \) |
| 97 | \( 1 + 3.17iT - 97T^{2} \) |
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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
−8.618224356114750721220184620723, −7.925436395302052315304191663009, −7.01148342538037096952642753797, −6.46266711702532417487448729415, −5.54048556802976233128648512309, −5.25254920105651099885189514911, −3.90345279784937682521860374131, −3.23334302545409057825073869560, −2.54403641019885675642643411216, −1.20457558474756277933388759389,
0.48404418645456168450557599645, 1.64352667013730545929929137312, 2.32477018535824715137719875716, 3.44659220496184176380521968067, 4.14795825929209062458366698872, 4.72411912330176563246726786870, 5.44960185031889371701694748888, 6.92585244243142722688176336706, 7.20048355841465874344377932823, 7.920209116975958282436046884870
