| L(s) = 1 | − 3-s + 4·7-s + 9-s + 4·13-s + 8·19-s − 4·21-s − 4·23-s − 27-s − 6·29-s + 8·31-s − 4·37-s − 4·39-s + 6·41-s − 4·43-s + 4·47-s + 9·49-s − 12·53-s − 8·57-s − 6·61-s + 4·63-s − 12·67-s + 4·69-s + 16·71-s + 8·79-s + 81-s + 12·83-s + 6·87-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.51·7-s + 1/3·9-s + 1.10·13-s + 1.83·19-s − 0.872·21-s − 0.834·23-s − 0.192·27-s − 1.11·29-s + 1.43·31-s − 0.657·37-s − 0.640·39-s + 0.937·41-s − 0.609·43-s + 0.583·47-s + 9/7·49-s − 1.64·53-s − 1.05·57-s − 0.768·61-s + 0.503·63-s − 1.46·67-s + 0.481·69-s + 1.89·71-s + 0.900·79-s + 1/9·81-s + 1.31·83-s + 0.643·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.982292377\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.982292377\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
| 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
−8.948760513198812884146314576672, −7.938034012155396642120596850234, −7.70007892913568568092010083657, −6.54905091387253005035442201825, −5.72548028621951047972031467144, −5.08342569893800361575337553961, −4.29009368499749123400757195822, −3.30653022132204983684757563201, −1.85966624953736986244513895085, −1.02018113746853668035397520877,
1.02018113746853668035397520877, 1.85966624953736986244513895085, 3.30653022132204983684757563201, 4.29009368499749123400757195822, 5.08342569893800361575337553961, 5.72548028621951047972031467144, 6.54905091387253005035442201825, 7.70007892913568568092010083657, 7.938034012155396642120596850234, 8.948760513198812884146314576672
