| L(s) = 1 | + i·2-s + i·3-s − 4-s − 6-s + 3i·7-s − i·8-s + 2·9-s + 11-s − i·12-s + 6i·13-s − 3·14-s + 16-s − 7i·17-s + 2i·18-s − 5·19-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s + 0.577i·3-s − 0.5·4-s − 0.408·6-s + 1.13i·7-s − 0.353i·8-s + 0.666·9-s + 0.301·11-s − 0.288i·12-s + 1.66i·13-s − 0.801·14-s + 0.250·16-s − 1.69i·17-s + 0.471i·18-s − 1.14·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.294166 + 1.24610i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.294166 + 1.24610i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 - iT - 3T^{2} \) |
| 7 | \( 1 - 3iT - 7T^{2} \) |
| 13 | \( 1 - 6iT - 13T^{2} \) |
| 17 | \( 1 + 7iT - 17T^{2} \) |
| 19 | \( 1 + 5T + 19T^{2} \) |
| 23 | \( 1 - 6iT - 23T^{2} \) |
| 29 | \( 1 + 5T + 29T^{2} \) |
| 31 | \( 1 + 3T + 31T^{2} \) |
| 37 | \( 1 - 3iT - 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 + 2iT - 47T^{2} \) |
| 53 | \( 1 - iT - 53T^{2} \) |
| 59 | \( 1 - 10T + 59T^{2} \) |
| 61 | \( 1 - 7T + 61T^{2} \) |
| 67 | \( 1 - 8iT - 67T^{2} \) |
| 71 | \( 1 - 7T + 71T^{2} \) |
| 73 | \( 1 + 14iT - 73T^{2} \) |
| 79 | \( 1 + 10T + 79T^{2} \) |
| 83 | \( 1 - 6iT - 83T^{2} \) |
| 89 | \( 1 - 15T + 89T^{2} \) |
| 97 | \( 1 + 12iT - 97T^{2} \) |
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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
−11.30507882954410098538484757380, −9.924985562807607904183772242187, −9.216441747453579076338936149054, −8.833590833937272714194167185340, −7.39209812944101491682247509674, −6.71538491869990097464655362134, −5.57241688825574351725069640204, −4.70587621601681675417897440204, −3.75007429883438060249741022758, −2.04599276658663454121497080206,
0.76887550019713147419647615256, 2.06462701055061654963223705148, 3.65203058309269662712530383262, 4.36770169222068280176371463352, 5.84280563667207028487462762207, 6.85487389309921397567274826896, 7.85492549886027616160729448087, 8.505081785704659628194314697391, 9.901084917741441723928456229539, 10.54511783684765172753483428333
