| L(s) = 1 | + 1.72·2-s + 2.56·3-s + 0.962·4-s + 2.01·5-s + 4.41·6-s + 7-s − 1.78·8-s + 3.57·9-s + 3.46·10-s + 3.24·11-s + 2.46·12-s − 2.61·13-s + 1.72·14-s + 5.16·15-s − 4.99·16-s − 1.03·17-s + 6.15·18-s − 7.18·19-s + 1.94·20-s + 2.56·21-s + 5.59·22-s − 8.35·23-s − 4.57·24-s − 0.939·25-s − 4.50·26-s + 1.48·27-s + 0.962·28-s + ⋯ |
| L(s) = 1 | + 1.21·2-s + 1.48·3-s + 0.481·4-s + 0.901·5-s + 1.80·6-s + 0.377·7-s − 0.631·8-s + 1.19·9-s + 1.09·10-s + 0.979·11-s + 0.712·12-s − 0.726·13-s + 0.460·14-s + 1.33·15-s − 1.24·16-s − 0.251·17-s + 1.45·18-s − 1.64·19-s + 0.433·20-s + 0.559·21-s + 1.19·22-s − 1.74·23-s − 0.934·24-s − 0.187·25-s − 0.883·26-s + 0.285·27-s + 0.181·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 889 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 889 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.834780257\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.834780257\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - T \) |
| 127 | \( 1 + T \) |
| good | 2 | \( 1 - 1.72T + 2T^{2} \) |
| 3 | \( 1 - 2.56T + 3T^{2} \) |
| 5 | \( 1 - 2.01T + 5T^{2} \) |
| 11 | \( 1 - 3.24T + 11T^{2} \) |
| 13 | \( 1 + 2.61T + 13T^{2} \) |
| 17 | \( 1 + 1.03T + 17T^{2} \) |
| 19 | \( 1 + 7.18T + 19T^{2} \) |
| 23 | \( 1 + 8.35T + 23T^{2} \) |
| 29 | \( 1 + 2.77T + 29T^{2} \) |
| 31 | \( 1 - 7.77T + 31T^{2} \) |
| 37 | \( 1 + 0.894T + 37T^{2} \) |
| 41 | \( 1 - 9.16T + 41T^{2} \) |
| 43 | \( 1 - 11.1T + 43T^{2} \) |
| 47 | \( 1 - 11.4T + 47T^{2} \) |
| 53 | \( 1 + 1.21T + 53T^{2} \) |
| 59 | \( 1 + 10.8T + 59T^{2} \) |
| 61 | \( 1 - 1.59T + 61T^{2} \) |
| 67 | \( 1 + 6.57T + 67T^{2} \) |
| 71 | \( 1 - 5.66T + 71T^{2} \) |
| 73 | \( 1 - 16.1T + 73T^{2} \) |
| 79 | \( 1 - 6.54T + 79T^{2} \) |
| 83 | \( 1 + 15.6T + 83T^{2} \) |
| 89 | \( 1 + 2.20T + 89T^{2} \) |
| 97 | \( 1 + 17.3T + 97T^{2} \) |
| 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
−9.844767628054766341287411046566, −9.262929799094324404288032694951, −8.559956463923774911861150168778, −7.62266878879091051554755339549, −6.40230319005915964540680317877, −5.78245171089964385196249851009, −4.33032920173460037318776313475, −4.02180344464972282887405473455, −2.58679534125138147292818821723, −2.05626571970177968909955836281,
2.05626571970177968909955836281, 2.58679534125138147292818821723, 4.02180344464972282887405473455, 4.33032920173460037318776313475, 5.78245171089964385196249851009, 6.40230319005915964540680317877, 7.62266878879091051554755339549, 8.559956463923774911861150168778, 9.262929799094324404288032694951, 9.844767628054766341287411046566
