| L(s) = 1 | − i·2-s + 2i·3-s − 4-s + 2·6-s − 4i·7-s + i·8-s − 9-s − 2i·12-s − 2i·13-s − 4·14-s + 16-s + i·18-s − 5·19-s + 8·21-s − 3i·23-s − 2·24-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + 1.15i·3-s − 0.5·4-s + 0.816·6-s − 1.51i·7-s + 0.353i·8-s − 0.333·9-s − 0.577i·12-s − 0.554i·13-s − 1.06·14-s + 0.250·16-s + 0.235i·18-s − 1.14·19-s + 1.74·21-s − 0.625i·23-s − 0.408·24-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6709668934\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6709668934\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 5 | \( 1 \) |
| 37 | \( 1 - iT \) |
| good | 3 | \( 1 - 2iT - 3T^{2} \) |
| 7 | \( 1 + 4iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 + 5T + 19T^{2} \) |
| 23 | \( 1 + 3iT - 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 41 | \( 1 + 9T + 41T^{2} \) |
| 43 | \( 1 - 7iT - 43T^{2} \) |
| 47 | \( 1 + 6iT - 47T^{2} \) |
| 53 | \( 1 + 9iT - 53T^{2} \) |
| 59 | \( 1 + 3T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 - 2iT - 67T^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 + 11iT - 73T^{2} \) |
| 79 | \( 1 - T + 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + 4iT - 97T^{2} \) |
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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
−9.157195120909342404331351925024, −8.329579133822585631240914571566, −7.43505913034175945579538266626, −6.52929939698704454154228845539, −5.28654261823079878659807843949, −4.51521152922079748874197953419, −3.88920906839122923584295958397, −3.23676376814266698981634121793, −1.74185190374422869639987143480, −0.23854859570729402197916925098,
1.64403715407506077159195231257, 2.42195894308291509913112852022, 3.80994341827620931734010713924, 5.00498368210930096604570625141, 5.85228592662468041193903672923, 6.38688263327356857258586394167, 7.21424440313567032400712830517, 7.86158113604315561757720093012, 8.797480707103018515621426829546, 9.090136437247485724288843429941
