| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s − 2·7-s + 8-s + 9-s + 3·11-s − 12-s − 2·13-s − 2·14-s + 16-s − 6·17-s + 18-s + 19-s + 2·21-s + 3·22-s − 3·23-s − 24-s − 2·26-s − 27-s − 2·28-s + 3·29-s − 7·31-s + 32-s − 3·33-s − 6·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.755·7-s + 0.353·8-s + 1/3·9-s + 0.904·11-s − 0.288·12-s − 0.554·13-s − 0.534·14-s + 1/4·16-s − 1.45·17-s + 0.235·18-s + 0.229·19-s + 0.436·21-s + 0.639·22-s − 0.625·23-s − 0.204·24-s − 0.392·26-s − 0.192·27-s − 0.377·28-s + 0.557·29-s − 1.25·31-s + 0.176·32-s − 0.522·33-s − 1.02·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - T + p T^{2} \) |
| 79 | \( 1 + 13 T + p T^{2} \) |
| 83 | \( 1 + 15 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−8.433477856934587280813895193453, −7.22606228216335578929928124987, −6.78126563333461869182342722100, −6.12006628241937071496287928653, −5.33308852889797180437620377846, −4.41672343231661974558373309437, −3.79755406766830018597678644255, −2.72810890379120769802109991139, −1.63709419436419940250475884004, 0,
1.63709419436419940250475884004, 2.72810890379120769802109991139, 3.79755406766830018597678644255, 4.41672343231661974558373309437, 5.33308852889797180437620377846, 6.12006628241937071496287928653, 6.78126563333461869182342722100, 7.22606228216335578929928124987, 8.433477856934587280813895193453
