| L(s) = 1 | + 3-s − 5-s + 7-s − 2·9-s − 3·11-s + 7·13-s − 15-s − 5·17-s + 4·19-s + 21-s + 4·23-s + 25-s − 5·27-s + 5·29-s + 2·31-s − 3·33-s − 35-s + 7·39-s + 6·41-s + 8·43-s + 2·45-s − 9·47-s + 49-s − 5·51-s + 3·55-s + 4·57-s + 12·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 0.377·7-s − 2/3·9-s − 0.904·11-s + 1.94·13-s − 0.258·15-s − 1.21·17-s + 0.917·19-s + 0.218·21-s + 0.834·23-s + 1/5·25-s − 0.962·27-s + 0.928·29-s + 0.359·31-s − 0.522·33-s − 0.169·35-s + 1.12·39-s + 0.937·41-s + 1.21·43-s + 0.298·45-s − 1.31·47-s + 1/7·49-s − 0.700·51-s + 0.404·55-s + 0.529·57-s + 1.56·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.016143368\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.016143368\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 7 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 16 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 9 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 3 T + p T^{2} \) |
| 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
−8.668098091186240301465957133246, −8.472758474206131892567641495130, −7.70509517442609287334023654055, −6.76087570958351408556308640215, −5.87787367149055605321045150974, −5.03833552565997461317853791236, −4.03807723835174391711734801829, −3.19392368709260422772099142876, −2.36591268566892625644480246684, −0.916090804667228214682453362701,
0.916090804667228214682453362701, 2.36591268566892625644480246684, 3.19392368709260422772099142876, 4.03807723835174391711734801829, 5.03833552565997461317853791236, 5.87787367149055605321045150974, 6.76087570958351408556308640215, 7.70509517442609287334023654055, 8.472758474206131892567641495130, 8.668098091186240301465957133246
