| L(s) = 1 | + 4·5-s + 7-s − 2·11-s + 6·13-s + 4·17-s − 4·19-s + 2·23-s + 11·25-s − 2·29-s + 4·35-s − 2·37-s − 4·43-s + 12·47-s + 49-s − 6·53-s − 8·55-s + 8·59-s − 6·61-s + 24·65-s − 8·67-s + 14·71-s − 2·73-s − 2·77-s − 12·79-s + 4·83-s + 16·85-s + 6·91-s + ⋯ |
| L(s) = 1 | + 1.78·5-s + 0.377·7-s − 0.603·11-s + 1.66·13-s + 0.970·17-s − 0.917·19-s + 0.417·23-s + 11/5·25-s − 0.371·29-s + 0.676·35-s − 0.328·37-s − 0.609·43-s + 1.75·47-s + 1/7·49-s − 0.824·53-s − 1.07·55-s + 1.04·59-s − 0.768·61-s + 2.97·65-s − 0.977·67-s + 1.66·71-s − 0.234·73-s − 0.227·77-s − 1.35·79-s + 0.439·83-s + 1.73·85-s + 0.628·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.176047511\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.176047511\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 5 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 14 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−8.621234568783701478950617886590, −7.79195451575718587485184031739, −6.78902576466258518849814546939, −6.06564976659535992293108524760, −5.62370507273544228914734043068, −4.91059247592313575518339819369, −3.78443484852293199292674683103, −2.79378519992244683258213909080, −1.88896973398279373978653204444, −1.12644399783657185408249656998,
1.12644399783657185408249656998, 1.88896973398279373978653204444, 2.79378519992244683258213909080, 3.78443484852293199292674683103, 4.91059247592313575518339819369, 5.62370507273544228914734043068, 6.06564976659535992293108524760, 6.78902576466258518849814546939, 7.79195451575718587485184031739, 8.621234568783701478950617886590
