| L(s) = 1 | + 3-s + 2·5-s + 9-s + 2·11-s + 2·15-s + 2·17-s − 2·23-s − 25-s + 27-s + 6·29-s + 4·31-s + 2·33-s + 6·37-s − 2·41-s + 2·45-s + 2·51-s − 6·53-s + 4·55-s − 12·59-s + 12·61-s + 12·67-s − 2·69-s + 10·71-s + 12·73-s − 75-s − 12·79-s + 81-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.894·5-s + 1/3·9-s + 0.603·11-s + 0.516·15-s + 0.485·17-s − 0.417·23-s − 1/5·25-s + 0.192·27-s + 1.11·29-s + 0.718·31-s + 0.348·33-s + 0.986·37-s − 0.312·41-s + 0.298·45-s + 0.280·51-s − 0.824·53-s + 0.539·55-s − 1.56·59-s + 1.53·61-s + 1.46·67-s − 0.240·69-s + 1.18·71-s + 1.40·73-s − 0.115·75-s − 1.35·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4704 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.222615829\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.222615829\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 - 12 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
| 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
−8.286888431596555286657334298365, −7.73854178489723702308111376486, −6.69894450238348765055660509469, −6.25757774983507607022752153906, −5.39842320766302478198171451779, −4.54420676214985832644687837204, −3.70337928799270777592208975099, −2.79368202791990232736895950901, −1.98810207865713462948946984742, −1.02538988745536652107482241574,
1.02538988745536652107482241574, 1.98810207865713462948946984742, 2.79368202791990232736895950901, 3.70337928799270777592208975099, 4.54420676214985832644687837204, 5.39842320766302478198171451779, 6.25757774983507607022752153906, 6.69894450238348765055660509469, 7.73854178489723702308111376486, 8.286888431596555286657334298365
