| L(s) = 1 | + 0.310·2-s + 3-s − 1.90·4-s + 2.19·5-s + 0.310·6-s − 3.79·7-s − 1.21·8-s + 9-s + 0.683·10-s + 4.23·11-s − 1.90·12-s − 13-s − 1.17·14-s + 2.19·15-s + 3.42·16-s − 4.06·17-s + 0.310·18-s + 6.04·19-s − 4.18·20-s − 3.79·21-s + 1.31·22-s + 3.95·23-s − 1.21·24-s − 0.161·25-s − 0.310·26-s + 27-s + 7.22·28-s + ⋯ |
| L(s) = 1 | + 0.219·2-s + 0.577·3-s − 0.951·4-s + 0.983·5-s + 0.126·6-s − 1.43·7-s − 0.429·8-s + 0.333·9-s + 0.216·10-s + 1.27·11-s − 0.549·12-s − 0.277·13-s − 0.315·14-s + 0.567·15-s + 0.857·16-s − 0.986·17-s + 0.0732·18-s + 1.38·19-s − 0.936·20-s − 0.828·21-s + 0.280·22-s + 0.825·23-s − 0.247·24-s − 0.0322·25-s − 0.0609·26-s + 0.192·27-s + 1.36·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.169323849\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.169323849\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 + T \) |
| good | 2 | \( 1 - 0.310T + 2T^{2} \) |
| 5 | \( 1 - 2.19T + 5T^{2} \) |
| 7 | \( 1 + 3.79T + 7T^{2} \) |
| 11 | \( 1 - 4.23T + 11T^{2} \) |
| 17 | \( 1 + 4.06T + 17T^{2} \) |
| 19 | \( 1 - 6.04T + 19T^{2} \) |
| 23 | \( 1 - 3.95T + 23T^{2} \) |
| 29 | \( 1 + 4.42T + 29T^{2} \) |
| 31 | \( 1 + 9.06T + 31T^{2} \) |
| 37 | \( 1 - 9.85T + 37T^{2} \) |
| 41 | \( 1 - 8.31T + 41T^{2} \) |
| 43 | \( 1 + 8.29T + 43T^{2} \) |
| 47 | \( 1 - 7.23T + 47T^{2} \) |
| 53 | \( 1 - 11.3T + 53T^{2} \) |
| 59 | \( 1 + 6.01T + 59T^{2} \) |
| 61 | \( 1 + 6.82T + 61T^{2} \) |
| 67 | \( 1 - 1.93T + 67T^{2} \) |
| 71 | \( 1 - 13.0T + 71T^{2} \) |
| 73 | \( 1 + 6.80T + 73T^{2} \) |
| 79 | \( 1 + 11.7T + 79T^{2} \) |
| 83 | \( 1 - 16.1T + 83T^{2} \) |
| 89 | \( 1 - 7.75T + 89T^{2} \) |
| 97 | \( 1 - 10.3T + 97T^{2} \) |
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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
−8.995367696586171101423798238635, −7.62507543517030636748716357298, −6.94680985551036568180616988290, −6.10221324222260155137935826802, −5.60906161751395389473829097928, −4.55815720974618426316727049801, −3.72392460917306217764072720951, −3.15610656571416273434859231593, −2.08696991102809422975001087437, −0.805211883306911368113415497980,
0.805211883306911368113415497980, 2.08696991102809422975001087437, 3.15610656571416273434859231593, 3.72392460917306217764072720951, 4.55815720974618426316727049801, 5.60906161751395389473829097928, 6.10221324222260155137935826802, 6.94680985551036568180616988290, 7.62507543517030636748716357298, 8.995367696586171101423798238635
