| L(s) = 1 | − 2-s − 1.73·3-s + 4-s + 1.73·6-s − 8-s + 1.99·9-s + 1.73·11-s − 1.73·12-s + 13-s + 16-s + 17-s − 1.99·18-s − 1.73·22-s + 1.73·24-s + 25-s − 26-s − 1.73·27-s − 32-s − 2.99·33-s − 34-s + 1.99·36-s − 1.73·39-s + 1.73·44-s − 1.73·48-s − 50-s − 1.73·51-s + 52-s + ⋯ |
| L(s) = 1 | − 2-s − 1.73·3-s + 4-s + 1.73·6-s − 8-s + 1.99·9-s + 1.73·11-s − 1.73·12-s + 13-s + 16-s + 17-s − 1.99·18-s − 1.73·22-s + 1.73·24-s + 25-s − 26-s − 1.73·27-s − 32-s − 2.99·33-s − 34-s + 1.99·36-s − 1.73·39-s + 1.73·44-s − 1.73·48-s − 50-s − 1.73·51-s + 52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5741399158\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5741399158\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 7 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 3 | \( 1 + 1.73T + T^{2} \) |
| 5 | \( 1 - T^{2} \) |
| 11 | \( 1 - 1.73T + T^{2} \) |
| 13 | \( 1 - T + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + T + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + 1.73T + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - 1.73T + T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - T + T^{2} \) |
| 97 | \( 1 - T^{2} \) |
| 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
−8.973293562638251319724270053544, −8.056522561671168512024461503494, −7.10350550302749743541646205977, −6.52341598717714413264105181402, −6.07660500680430133180969592635, −5.31004788033584759314288198884, −4.21815871410334199376265175633, −3.29416024314644885509815301305, −1.54204151387104326027237222343, −0.960671531228963139956870116350,
0.960671531228963139956870116350, 1.54204151387104326027237222343, 3.29416024314644885509815301305, 4.21815871410334199376265175633, 5.31004788033584759314288198884, 6.07660500680430133180969592635, 6.52341598717714413264105181402, 7.10350550302749743541646205977, 8.056522561671168512024461503494, 8.973293562638251319724270053544
