| L(s) = 1 | + i·2-s − 1.81i·3-s − 4-s + 1.81·6-s + 2.52i·7-s − i·8-s − 0.289·9-s − 2.91·11-s + 1.81i·12-s − 3.10i·13-s − 2.52·14-s + 16-s + 6.91i·17-s − 0.289i·18-s − 1.81·19-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 1.04i·3-s − 0.5·4-s + 0.740·6-s + 0.954i·7-s − 0.353i·8-s − 0.0963·9-s − 0.879·11-s + 0.523i·12-s − 0.860i·13-s − 0.674·14-s + 0.250·16-s + 1.67i·17-s − 0.0681i·18-s − 0.416·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9572709439\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9572709439\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 29 | \( 1 + T \) |
| good | 3 | \( 1 + 1.81iT - 3T^{2} \) |
| 7 | \( 1 - 2.52iT - 7T^{2} \) |
| 11 | \( 1 + 2.91T + 11T^{2} \) |
| 13 | \( 1 + 3.10iT - 13T^{2} \) |
| 17 | \( 1 - 6.91iT - 17T^{2} \) |
| 19 | \( 1 + 1.81T + 19T^{2} \) |
| 23 | \( 1 - 5.68iT - 23T^{2} \) |
| 31 | \( 1 + 8.12T + 31T^{2} \) |
| 37 | \( 1 - 7.39iT - 37T^{2} \) |
| 41 | \( 1 - 8.39T + 41T^{2} \) |
| 43 | \( 1 - 5.04iT - 43T^{2} \) |
| 47 | \( 1 - 4.86iT - 47T^{2} \) |
| 53 | \( 1 + 5.30iT - 53T^{2} \) |
| 59 | \( 1 + 2.60T + 59T^{2} \) |
| 61 | \( 1 + 10.0T + 61T^{2} \) |
| 67 | \( 1 + 2.68iT - 67T^{2} \) |
| 71 | \( 1 - 5.68T + 71T^{2} \) |
| 73 | \( 1 - 15.0iT - 73T^{2} \) |
| 79 | \( 1 - 1.73T + 79T^{2} \) |
| 83 | \( 1 - 13.1iT - 83T^{2} \) |
| 89 | \( 1 + 5.23T + 89T^{2} \) |
| 97 | \( 1 - 1.27iT - 97T^{2} \) |
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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
−9.609576519116372420367763699859, −8.667050610870465982218922490839, −7.916300059972051845517650826244, −7.58529621187963598399165904158, −6.43004719471931714301924301241, −5.86200352116649791482951212561, −5.15241276373428237274104172890, −3.82046933689122017144405633137, −2.56616887274346185010583350154, −1.44703607625611833275518825717,
0.38654334699099946757716324355, 2.11848267915247722735737454602, 3.24984839759305665718932998786, 4.23696167485628473193703507935, 4.65488623840619821175676546302, 5.62726615900960545825201748958, 7.04506911546536444296312755745, 7.59390836406854768304524452899, 8.961155083761643890626592292569, 9.314876116518495693649763379627
