| L(s) = 1 | − 3-s + 5-s + 2·7-s + 9-s + 4·11-s − 13-s − 15-s + 2·19-s − 2·21-s − 2·23-s + 25-s − 27-s + 4·29-s + 4·31-s − 4·33-s + 2·35-s − 2·37-s + 39-s − 6·41-s − 4·43-s + 45-s + 8·47-s − 3·49-s − 2·53-s + 4·55-s − 2·57-s + 4·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 0.755·7-s + 1/3·9-s + 1.20·11-s − 0.277·13-s − 0.258·15-s + 0.458·19-s − 0.436·21-s − 0.417·23-s + 1/5·25-s − 0.192·27-s + 0.742·29-s + 0.718·31-s − 0.696·33-s + 0.338·35-s − 0.328·37-s + 0.160·39-s − 0.937·41-s − 0.609·43-s + 0.149·45-s + 1.16·47-s − 3/7·49-s − 0.274·53-s + 0.539·55-s − 0.264·57-s + 0.520·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.810813838\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.810813838\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
| 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
−9.520009134295800655907776929717, −8.676563824242227720907246514036, −7.84347320477430312279068906121, −6.83753672766789611040877540710, −6.25779845443492060914021328840, −5.25787500780889329278664829328, −4.58917803356227598993084088957, −3.53655890135222970284137728225, −2.09771084382809545410112476360, −1.05434995902020695342243240802,
1.05434995902020695342243240802, 2.09771084382809545410112476360, 3.53655890135222970284137728225, 4.58917803356227598993084088957, 5.25787500780889329278664829328, 6.25779845443492060914021328840, 6.83753672766789611040877540710, 7.84347320477430312279068906121, 8.676563824242227720907246514036, 9.520009134295800655907776929717
