| L(s) = 1 | + 3·3-s + 5·5-s − 7·7-s + 9·9-s − 12·11-s + 30·13-s + 15·15-s − 134·17-s + 92·19-s − 21·21-s − 112·23-s + 25·25-s + 27·27-s − 58·29-s + 224·31-s − 36·33-s − 35·35-s − 146·37-s + 90·39-s + 18·41-s − 340·43-s + 45·45-s − 208·47-s + 49·49-s − 402·51-s − 754·53-s − 60·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 0.377·7-s + 1/3·9-s − 0.328·11-s + 0.640·13-s + 0.258·15-s − 1.91·17-s + 1.11·19-s − 0.218·21-s − 1.01·23-s + 1/5·25-s + 0.192·27-s − 0.371·29-s + 1.29·31-s − 0.189·33-s − 0.169·35-s − 0.648·37-s + 0.369·39-s + 0.0685·41-s − 1.20·43-s + 0.149·45-s − 0.645·47-s + 1/7·49-s − 1.10·51-s − 1.95·53-s − 0.147·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p T \) |
| 5 | \( 1 - p T \) |
| 7 | \( 1 + p T \) |
| good | 11 | \( 1 + 12 T + p^{3} T^{2} \) |
| 13 | \( 1 - 30 T + p^{3} T^{2} \) |
| 17 | \( 1 + 134 T + p^{3} T^{2} \) |
| 19 | \( 1 - 92 T + p^{3} T^{2} \) |
| 23 | \( 1 + 112 T + p^{3} T^{2} \) |
| 29 | \( 1 + 2 p T + p^{3} T^{2} \) |
| 31 | \( 1 - 224 T + p^{3} T^{2} \) |
| 37 | \( 1 + 146 T + p^{3} T^{2} \) |
| 41 | \( 1 - 18 T + p^{3} T^{2} \) |
| 43 | \( 1 + 340 T + p^{3} T^{2} \) |
| 47 | \( 1 + 208 T + p^{3} T^{2} \) |
| 53 | \( 1 + 754 T + p^{3} T^{2} \) |
| 59 | \( 1 + 380 T + p^{3} T^{2} \) |
| 61 | \( 1 - 718 T + p^{3} T^{2} \) |
| 67 | \( 1 + 412 T + p^{3} T^{2} \) |
| 71 | \( 1 - 960 T + p^{3} T^{2} \) |
| 73 | \( 1 - 1066 T + p^{3} T^{2} \) |
| 79 | \( 1 + 896 T + p^{3} T^{2} \) |
| 83 | \( 1 + 436 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1038 T + p^{3} T^{2} \) |
| 97 | \( 1 + 702 T + p^{3} T^{2} \) |
| 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
−8.543481349741917636047345990590, −8.010990647727093230688422972333, −6.85331684118774639790377385827, −6.35881672344508460526435650590, −5.29009273179942624697250231256, −4.35103005648574436351837721012, −3.37514022408927255250730284216, −2.46379457244669572329142236479, −1.49755028586111857837268868380, 0,
1.49755028586111857837268868380, 2.46379457244669572329142236479, 3.37514022408927255250730284216, 4.35103005648574436351837721012, 5.29009273179942624697250231256, 6.35881672344508460526435650590, 6.85331684118774639790377385827, 8.010990647727093230688422972333, 8.543481349741917636047345990590
