| L(s) = 1 | − 0.719·2-s − 3-s − 1.48·4-s − 1.74·5-s + 0.719·6-s − 7-s + 2.50·8-s + 9-s + 1.25·10-s − 3.78·11-s + 1.48·12-s + 4.32·13-s + 0.719·14-s + 1.74·15-s + 1.16·16-s − 3.44·17-s − 0.719·18-s − 6.94·19-s + 2.58·20-s + 21-s + 2.72·22-s + 1.20·23-s − 2.50·24-s − 1.96·25-s − 3.10·26-s − 27-s + 1.48·28-s + ⋯ |
| L(s) = 1 | − 0.508·2-s − 0.577·3-s − 0.741·4-s − 0.779·5-s + 0.293·6-s − 0.377·7-s + 0.885·8-s + 0.333·9-s + 0.396·10-s − 1.14·11-s + 0.428·12-s + 1.19·13-s + 0.192·14-s + 0.449·15-s + 0.291·16-s − 0.834·17-s − 0.169·18-s − 1.59·19-s + 0.577·20-s + 0.218·21-s + 0.580·22-s + 0.251·23-s − 0.511·24-s − 0.392·25-s − 0.609·26-s − 0.192·27-s + 0.280·28-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
| \(L(1)\) |
\(\approx\) |
\(0.2165101452\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2165101452\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.719T + 2T^{2} \) |
| 5 | \( 1 + 1.74T + 5T^{2} \) |
| 11 | \( 1 + 3.78T + 11T^{2} \) |
| 13 | \( 1 - 4.32T + 13T^{2} \) |
| 17 | \( 1 + 3.44T + 17T^{2} \) |
| 19 | \( 1 + 6.94T + 19T^{2} \) |
| 23 | \( 1 - 1.20T + 23T^{2} \) |
| 29 | \( 1 - 9.18T + 29T^{2} \) |
| 31 | \( 1 + 0.218T + 31T^{2} \) |
| 37 | \( 1 + 1.80T + 37T^{2} \) |
| 41 | \( 1 + 0.597T + 41T^{2} \) |
| 43 | \( 1 + 4.58T + 43T^{2} \) |
| 47 | \( 1 + 3.05T + 47T^{2} \) |
| 53 | \( 1 - 2.40T + 53T^{2} \) |
| 59 | \( 1 + 7.84T + 59T^{2} \) |
| 61 | \( 1 - 12.7T + 61T^{2} \) |
| 67 | \( 1 - 2.19T + 67T^{2} \) |
| 71 | \( 1 + 12.8T + 71T^{2} \) |
| 73 | \( 1 + 3.36T + 73T^{2} \) |
| 79 | \( 1 + 16.5T + 79T^{2} \) |
| 83 | \( 1 + 8.69T + 83T^{2} \) |
| 89 | \( 1 - 0.0226T + 89T^{2} \) |
| 97 | \( 1 - 3.32T + 97T^{2} \) |
| 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
−8.054874750014958021764469552306, −7.18619691146671621591672646735, −6.49927007878173939378069893107, −5.79846058457183783671434279317, −4.89536512501851983736201624308, −4.33458446418182053840465827474, −3.72763825577585119502414401833, −2.66935219074018513721363644525, −1.44242125287377623184005277654, −0.27021821200378390944506884525,
0.27021821200378390944506884525, 1.44242125287377623184005277654, 2.66935219074018513721363644525, 3.72763825577585119502414401833, 4.33458446418182053840465827474, 4.89536512501851983736201624308, 5.79846058457183783671434279317, 6.49927007878173939378069893107, 7.18619691146671621591672646735, 8.054874750014958021764469552306
