| L(s) = 1 | − 4.38·2-s − 0.971·3-s + 11.2·4-s − 20.4·5-s + 4.26·6-s − 7·7-s − 14.1·8-s − 26.0·9-s + 89.6·10-s − 33.3·11-s − 10.9·12-s + 3.96·13-s + 30.6·14-s + 19.8·15-s − 27.6·16-s − 57.5·17-s + 114.·18-s − 43.1·19-s − 229.·20-s + 6.80·21-s + 146.·22-s − 23·23-s + 13.7·24-s + 292.·25-s − 17.3·26-s + 51.5·27-s − 78.6·28-s + ⋯ |
| L(s) = 1 | − 1.55·2-s − 0.187·3-s + 1.40·4-s − 1.82·5-s + 0.289·6-s − 0.377·7-s − 0.626·8-s − 0.965·9-s + 2.83·10-s − 0.915·11-s − 0.262·12-s + 0.0845·13-s + 0.586·14-s + 0.341·15-s − 0.432·16-s − 0.820·17-s + 1.49·18-s − 0.521·19-s − 2.56·20-s + 0.0706·21-s + 1.41·22-s − 0.208·23-s + 0.117·24-s + 2.34·25-s − 0.131·26-s + 0.367·27-s − 0.530·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.1651673733\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1651673733\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + 7T \) |
| 23 | \( 1 + 23T \) |
| good | 2 | \( 1 + 4.38T + 8T^{2} \) |
| 3 | \( 1 + 0.971T + 27T^{2} \) |
| 5 | \( 1 + 20.4T + 125T^{2} \) |
| 11 | \( 1 + 33.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 3.96T + 2.19e3T^{2} \) |
| 17 | \( 1 + 57.5T + 4.91e3T^{2} \) |
| 19 | \( 1 + 43.1T + 6.85e3T^{2} \) |
| 29 | \( 1 + 51.1T + 2.43e4T^{2} \) |
| 31 | \( 1 - 241.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 103.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 478.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 127.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 281.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 526.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 133.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 125.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 361.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 651.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 627.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 589.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 429.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.00e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.05e3T + 9.12e5T^{2} \) |
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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
−11.90717430750861529820826772065, −11.18461671619729079408303062028, −10.51312101767434359221207698949, −9.103064064034386760472251098711, −8.219941865108622884635994595074, −7.67816830944257498138638990932, −6.44755252674358362557995748845, −4.50288912145419821407148180338, −2.83127730106000629944903835708, −0.38787903110878489742073558137,
0.38787903110878489742073558137, 2.83127730106000629944903835708, 4.50288912145419821407148180338, 6.44755252674358362557995748845, 7.67816830944257498138638990932, 8.219941865108622884635994595074, 9.103064064034386760472251098711, 10.51312101767434359221207698949, 11.18461671619729079408303062028, 11.90717430750861529820826772065
