| L(s) = 1 | − 0.607·2-s − 3-s − 1.63·4-s − 3.70·5-s + 0.607·6-s + 0.175·7-s + 2.20·8-s + 9-s + 2.24·10-s − 2.41·11-s + 1.63·12-s − 13-s − 0.106·14-s + 3.70·15-s + 1.92·16-s + 1.06·17-s − 0.607·18-s − 7.45·19-s + 6.03·20-s − 0.175·21-s + 1.46·22-s − 1.86·23-s − 2.20·24-s + 8.69·25-s + 0.607·26-s − 27-s − 0.285·28-s + ⋯ |
| L(s) = 1 | − 0.429·2-s − 0.577·3-s − 0.815·4-s − 1.65·5-s + 0.248·6-s + 0.0662·7-s + 0.780·8-s + 0.333·9-s + 0.711·10-s − 0.726·11-s + 0.470·12-s − 0.277·13-s − 0.0284·14-s + 0.955·15-s + 0.480·16-s + 0.258·17-s − 0.143·18-s − 1.71·19-s + 1.34·20-s − 0.0382·21-s + 0.312·22-s − 0.388·23-s − 0.450·24-s + 1.73·25-s + 0.119·26-s − 0.192·27-s − 0.0540·28-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
| \(L(1)\) |
\(\approx\) |
\(0.02007604427\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.02007604427\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.607T + 2T^{2} \) |
| 5 | \( 1 + 3.70T + 5T^{2} \) |
| 7 | \( 1 - 0.175T + 7T^{2} \) |
| 11 | \( 1 + 2.41T + 11T^{2} \) |
| 17 | \( 1 - 1.06T + 17T^{2} \) |
| 19 | \( 1 + 7.45T + 19T^{2} \) |
| 23 | \( 1 + 1.86T + 23T^{2} \) |
| 29 | \( 1 - 4.19T + 29T^{2} \) |
| 31 | \( 1 + 9.41T + 31T^{2} \) |
| 37 | \( 1 + 1.36T + 37T^{2} \) |
| 41 | \( 1 + 11.3T + 41T^{2} \) |
| 43 | \( 1 + 3.16T + 43T^{2} \) |
| 47 | \( 1 + 0.0643T + 47T^{2} \) |
| 53 | \( 1 + 9.84T + 53T^{2} \) |
| 59 | \( 1 - 0.754T + 59T^{2} \) |
| 61 | \( 1 + 8.76T + 61T^{2} \) |
| 67 | \( 1 + 6.87T + 67T^{2} \) |
| 71 | \( 1 - 1.97T + 71T^{2} \) |
| 73 | \( 1 + 15.3T + 73T^{2} \) |
| 79 | \( 1 - 13.8T + 79T^{2} \) |
| 83 | \( 1 + 4.54T + 83T^{2} \) |
| 89 | \( 1 - 8.54T + 89T^{2} \) |
| 97 | \( 1 + 12.8T + 97T^{2} \) |
| 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
−8.325027880712356509967210932519, −7.87255645097539717501244289541, −7.21267830205426490279584532230, −6.34939071509432829986917413788, −5.20063439952844173870525705934, −4.66353340555188387247002722437, −4.00014109306747554606331134884, −3.20537173702997798772281621727, −1.65334559214061184486254567168, −0.089482427811235835095013378791,
0.089482427811235835095013378791, 1.65334559214061184486254567168, 3.20537173702997798772281621727, 4.00014109306747554606331134884, 4.66353340555188387247002722437, 5.20063439952844173870525705934, 6.34939071509432829986917413788, 7.21267830205426490279584532230, 7.87255645097539717501244289541, 8.325027880712356509967210932519
