L(s) = 1 | + 1.92·2-s + 3-s + 1.68·4-s − 3.20·5-s + 1.92·6-s − 7-s − 0.596·8-s + 9-s − 6.16·10-s − 2.87·11-s + 1.68·12-s − 1.24·13-s − 1.92·14-s − 3.20·15-s − 4.52·16-s − 0.348·17-s + 1.92·18-s + 0.401·19-s − 5.42·20-s − 21-s − 5.51·22-s − 0.203·23-s − 0.596·24-s + 5.29·25-s − 2.39·26-s + 27-s − 1.68·28-s + ⋯ |
L(s) = 1 | + 1.35·2-s + 0.577·3-s + 0.844·4-s − 1.43·5-s + 0.784·6-s − 0.377·7-s − 0.210·8-s + 0.333·9-s − 1.94·10-s − 0.866·11-s + 0.487·12-s − 0.345·13-s − 0.513·14-s − 0.828·15-s − 1.13·16-s − 0.0845·17-s + 0.452·18-s + 0.0921·19-s − 1.21·20-s − 0.218·21-s − 1.17·22-s − 0.0424·23-s − 0.121·24-s + 1.05·25-s − 0.469·26-s + 0.192·27-s − 0.319·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.645570984\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.645570984\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 383 | \( 1 + T \) |
good | 2 | \( 1 - 1.92T + 2T^{2} \) |
| 5 | \( 1 + 3.20T + 5T^{2} \) |
| 11 | \( 1 + 2.87T + 11T^{2} \) |
| 13 | \( 1 + 1.24T + 13T^{2} \) |
| 17 | \( 1 + 0.348T + 17T^{2} \) |
| 19 | \( 1 - 0.401T + 19T^{2} \) |
| 23 | \( 1 + 0.203T + 23T^{2} \) |
| 29 | \( 1 + 2.31T + 29T^{2} \) |
| 31 | \( 1 - 3.93T + 31T^{2} \) |
| 37 | \( 1 - 0.880T + 37T^{2} \) |
| 41 | \( 1 - 10.8T + 41T^{2} \) |
| 43 | \( 1 - 11.8T + 43T^{2} \) |
| 47 | \( 1 - 7.32T + 47T^{2} \) |
| 53 | \( 1 - 1.13T + 53T^{2} \) |
| 59 | \( 1 - 3.64T + 59T^{2} \) |
| 61 | \( 1 + 9.78T + 61T^{2} \) |
| 67 | \( 1 + 1.45T + 67T^{2} \) |
| 71 | \( 1 - 11.8T + 71T^{2} \) |
| 73 | \( 1 + 6.76T + 73T^{2} \) |
| 79 | \( 1 - 13.6T + 79T^{2} \) |
| 83 | \( 1 - 11.7T + 83T^{2} \) |
| 89 | \( 1 + 6.26T + 89T^{2} \) |
| 97 | \( 1 + 15.5T + 97T^{2} \) |
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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
−7.61108178608270481458499023572, −7.27409620694951579118564189856, −6.32218306197346786671754791084, −5.60175643167749962215613224666, −4.78347133136295707122014065562, −4.16403982471797526324986572363, −3.72067485011567630526108790343, −2.85831363019288552119842396241, −2.41077704090029502915419570811, −0.60193164864688110383420309370,
0.60193164864688110383420309370, 2.41077704090029502915419570811, 2.85831363019288552119842396241, 3.72067485011567630526108790343, 4.16403982471797526324986572363, 4.78347133136295707122014065562, 5.60175643167749962215613224666, 6.32218306197346786671754791084, 7.27409620694951579118564189856, 7.61108178608270481458499023572