| L(s) = 1 | + i·2-s + i·3-s − 4-s − 6-s + 2.73i·7-s − i·8-s − 9-s + 5.19·11-s − i·12-s − 4.73i·13-s − 2.73·14-s + 16-s − 2.73i·17-s − i·18-s − 19-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s + 0.577i·3-s − 0.5·4-s − 0.408·6-s + 1.03i·7-s − 0.353i·8-s − 0.333·9-s + 1.56·11-s − 0.288i·12-s − 1.31i·13-s − 0.730·14-s + 0.250·16-s − 0.662i·17-s − 0.235i·18-s − 0.229·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.676138125\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.676138125\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 7 | \( 1 - 2.73iT - 7T^{2} \) |
| 11 | \( 1 - 5.19T + 11T^{2} \) |
| 13 | \( 1 + 4.73iT - 13T^{2} \) |
| 17 | \( 1 + 2.73iT - 17T^{2} \) |
| 23 | \( 1 + 8.46iT - 23T^{2} \) |
| 29 | \( 1 - 9.19T + 29T^{2} \) |
| 31 | \( 1 + 5.92T + 31T^{2} \) |
| 37 | \( 1 + 8.92iT - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 8.73iT - 43T^{2} \) |
| 47 | \( 1 + 3.46iT - 47T^{2} \) |
| 53 | \( 1 - 4.66iT - 53T^{2} \) |
| 59 | \( 1 + 2.19T + 59T^{2} \) |
| 61 | \( 1 - 8.26T + 61T^{2} \) |
| 67 | \( 1 + 5.19iT - 67T^{2} \) |
| 71 | \( 1 + 10.1T + 71T^{2} \) |
| 73 | \( 1 + 14.4iT - 73T^{2} \) |
| 79 | \( 1 + 5.92T + 79T^{2} \) |
| 83 | \( 1 + 16.6iT - 83T^{2} \) |
| 89 | \( 1 + 3.92T + 89T^{2} \) |
| 97 | \( 1 - 17.1iT - 97T^{2} \) |
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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
−8.932206456675744030248363658269, −8.272170016665046572705031462605, −7.26320730903916352765370446056, −6.42835983486829777542521345306, −5.81578776710111285116947216257, −5.05888477595579435738330192363, −4.25772550500425859106122850222, −3.32865210633205192943050680409, −2.32929427138857601982223353164, −0.60821917671327297329401974600,
1.24293411009862054258275413861, 1.61154873939506403363401661826, 3.06469080175731566477482545873, 4.01248942756544603472994350905, 4.41054633348794843918156735344, 5.71966938848649403552891938964, 6.71360134348692321141321321070, 6.99720905638355235996166067790, 8.094518122497850664604975263296, 8.777384622170783526297593092846
