| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 1.73·7-s + 8-s + 9-s + 1.73·11-s + 12-s + 4·13-s + 1.73·14-s + 16-s − 3.46·17-s + 18-s + 1.73·21-s + 1.73·22-s + 24-s − 5·25-s + 4·26-s + 27-s + 1.73·28-s + 3·29-s + 7·31-s + 32-s + 1.73·33-s − 3.46·34-s + 36-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.408·6-s + 0.654·7-s + 0.353·8-s + 0.333·9-s + 0.522·11-s + 0.288·12-s + 1.10·13-s + 0.462·14-s + 0.250·16-s − 0.840·17-s + 0.235·18-s + 0.377·21-s + 0.369·22-s + 0.204·24-s − 25-s + 0.784·26-s + 0.192·27-s + 0.327·28-s + 0.557·29-s + 1.25·31-s + 0.176·32-s + 0.301·33-s − 0.594·34-s + 0.166·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3174 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3174 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.340676908\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.340676908\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 5 | \( 1 + 5T^{2} \) |
| 7 | \( 1 - 1.73T + 7T^{2} \) |
| 11 | \( 1 - 1.73T + 11T^{2} \) |
| 13 | \( 1 - 4T + 13T^{2} \) |
| 17 | \( 1 + 3.46T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 29 | \( 1 - 3T + 29T^{2} \) |
| 31 | \( 1 - 7T + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 - 12T + 41T^{2} \) |
| 43 | \( 1 + 10.3T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 8.66T + 53T^{2} \) |
| 59 | \( 1 - 9T + 59T^{2} \) |
| 61 | \( 1 + 6.92T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 + 11T + 73T^{2} \) |
| 79 | \( 1 - 5.19T + 79T^{2} \) |
| 83 | \( 1 + 1.73T + 83T^{2} \) |
| 89 | \( 1 + 10.3T + 89T^{2} \) |
| 97 | \( 1 - 15.5T + 97T^{2} \) |
| 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
−8.487127107700892455625944750421, −8.058924696964026036028443382179, −7.07864963132833544211141051169, −6.37358640444479522080927836729, −5.65675921960951512814730703112, −4.53936273533095992768000041935, −4.10406205637239148045407028212, −3.15293173428103939617301384509, −2.17474328382894297088108515876, −1.22987364841348953849682824094,
1.22987364841348953849682824094, 2.17474328382894297088108515876, 3.15293173428103939617301384509, 4.10406205637239148045407028212, 4.53936273533095992768000041935, 5.65675921960951512814730703112, 6.37358640444479522080927836729, 7.07864963132833544211141051169, 8.058924696964026036028443382179, 8.487127107700892455625944750421
