| L(s) = 1 | − 0.732·2-s − 3-s − 1.46·4-s − 5-s + 0.732·6-s + 7-s + 2.53·8-s + 9-s + 0.732·10-s − 11-s + 1.46·12-s − 4.19·13-s − 0.732·14-s + 15-s + 1.07·16-s − 6.46·17-s − 0.732·18-s − 1.53·19-s + 1.46·20-s − 21-s + 0.732·22-s + 8.46·23-s − 2.53·24-s + 25-s + 3.07·26-s − 27-s − 1.46·28-s + ⋯ |
| L(s) = 1 | − 0.517·2-s − 0.577·3-s − 0.732·4-s − 0.447·5-s + 0.298·6-s + 0.377·7-s + 0.896·8-s + 0.333·9-s + 0.231·10-s − 0.301·11-s + 0.422·12-s − 1.16·13-s − 0.195·14-s + 0.258·15-s + 0.267·16-s − 1.56·17-s − 0.172·18-s − 0.352·19-s + 0.327·20-s − 0.218·21-s + 0.156·22-s + 1.76·23-s − 0.517·24-s + 0.200·25-s + 0.602·26-s − 0.192·27-s − 0.276·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5574904557\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5574904557\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 + 0.732T + 2T^{2} \) |
| 13 | \( 1 + 4.19T + 13T^{2} \) |
| 17 | \( 1 + 6.46T + 17T^{2} \) |
| 19 | \( 1 + 1.53T + 19T^{2} \) |
| 23 | \( 1 - 8.46T + 23T^{2} \) |
| 29 | \( 1 - 5.73T + 29T^{2} \) |
| 31 | \( 1 + 5.26T + 31T^{2} \) |
| 37 | \( 1 + 4.19T + 37T^{2} \) |
| 41 | \( 1 + 7.66T + 41T^{2} \) |
| 43 | \( 1 - 1.73T + 43T^{2} \) |
| 47 | \( 1 - 1.26T + 47T^{2} \) |
| 53 | \( 1 - 4.46T + 53T^{2} \) |
| 59 | \( 1 - 13.7T + 59T^{2} \) |
| 61 | \( 1 - 11.3T + 61T^{2} \) |
| 67 | \( 1 - 4.92T + 67T^{2} \) |
| 71 | \( 1 - 7.66T + 71T^{2} \) |
| 73 | \( 1 - 8.39T + 73T^{2} \) |
| 79 | \( 1 - 3.66T + 79T^{2} \) |
| 83 | \( 1 - 2.46T + 83T^{2} \) |
| 89 | \( 1 - 14.6T + 89T^{2} \) |
| 97 | \( 1 + 2.26T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.790559447414366264959554886816, −8.884757785637890613151656278668, −8.353636075367501471609397000585, −7.27286628794809562431611589932, −6.76136701513419556854636607380, −5.11988230490427366285750778897, −4.89727294568226387950011100045, −3.81793811967262017526755387365, −2.24000732852800723036843551665, −0.61741691731344019653415010676,
0.61741691731344019653415010676, 2.24000732852800723036843551665, 3.81793811967262017526755387365, 4.89727294568226387950011100045, 5.11988230490427366285750778897, 6.76136701513419556854636607380, 7.27286628794809562431611589932, 8.353636075367501471609397000585, 8.884757785637890613151656278668, 9.790559447414366264959554886816
