| L(s) = 1 | − 0.618·3-s − 0.618·9-s + 1.61·17-s − 1.61·19-s + 25-s + 27-s − 0.618·41-s + 0.618·43-s + 49-s − 1.00·51-s + 1.00·57-s + 1.61·59-s + 1.61·67-s − 0.618·73-s − 0.618·75-s + 0.618·83-s + 0.618·89-s + 0.618·97-s + 0.618·107-s − 1.61·113-s + ⋯ |
| L(s) = 1 | − 0.618·3-s − 0.618·9-s + 1.61·17-s − 1.61·19-s + 25-s + 27-s − 0.618·41-s + 0.618·43-s + 49-s − 1.00·51-s + 1.00·57-s + 1.61·59-s + 1.61·67-s − 0.618·73-s − 0.618·75-s + 0.618·83-s + 0.618·89-s + 0.618·97-s + 0.618·107-s − 1.61·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9524572197\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9524572197\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + 0.618T + T^{2} \) |
| 5 | \( 1 - T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 - 1.61T + T^{2} \) |
| 19 | \( 1 + 1.61T + T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + 0.618T + T^{2} \) |
| 43 | \( 1 - 0.618T + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - 1.61T + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - 1.61T + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + 0.618T + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - 0.618T + T^{2} \) |
| 89 | \( 1 - 0.618T + T^{2} \) |
| 97 | \( 1 - 0.618T + T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.565647784370595190502790995946, −8.042606986030999960961282158581, −7.05755724839075784537319836846, −6.39432954959763976762670521869, −5.64364103041212903779460514456, −5.08759692084808761035575833580, −4.11234899924562735109260898428, −3.19561060457668301961683293087, −2.24128335441948430146323016105, −0.858483305858631633718486700969,
0.858483305858631633718486700969, 2.24128335441948430146323016105, 3.19561060457668301961683293087, 4.11234899924562735109260898428, 5.08759692084808761035575833580, 5.64364103041212903779460514456, 6.39432954959763976762670521869, 7.05755724839075784537319836846, 8.042606986030999960961282158581, 8.565647784370595190502790995946
