L(s) = 1 | + 2·3-s + 9-s − 3·11-s − 2·13-s + 4·17-s + 3·23-s − 4·27-s + 29-s + 2·31-s − 6·33-s − 7·37-s − 4·39-s − 2·41-s + 43-s − 12·47-s + 8·51-s − 6·53-s − 6·59-s − 6·61-s + 7·67-s + 6·69-s − 3·71-s − 2·73-s − 5·79-s − 11·81-s − 6·83-s + 2·87-s + ⋯ |
L(s) = 1 | + 1.15·3-s + 1/3·9-s − 0.904·11-s − 0.554·13-s + 0.970·17-s + 0.625·23-s − 0.769·27-s + 0.185·29-s + 0.359·31-s − 1.04·33-s − 1.15·37-s − 0.640·39-s − 0.312·41-s + 0.152·43-s − 1.75·47-s + 1.12·51-s − 0.824·53-s − 0.781·59-s − 0.768·61-s + 0.855·67-s + 0.722·69-s − 0.356·71-s − 0.234·73-s − 0.562·79-s − 1.22·81-s − 0.658·83-s + 0.214·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 5 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−7.52178636990314422865019492988, −6.85803050929673325271159715324, −5.92806096059354900363106753123, −5.16983003844481782775515812295, −4.59895590885912441240633132798, −3.45165944228311062525896211231, −3.09274966535921264912882201439, −2.33482497336505442896647392169, −1.44415190696755308120371946522, 0,
1.44415190696755308120371946522, 2.33482497336505442896647392169, 3.09274966535921264912882201439, 3.45165944228311062525896211231, 4.59895590885912441240633132798, 5.16983003844481782775515812295, 5.92806096059354900363106753123, 6.85803050929673325271159715324, 7.52178636990314422865019492988