| L(s) = 1 | − 0.102·2-s + 3.12·3-s − 1.98·4-s − 0.280·5-s − 0.319·6-s + 0.407·8-s + 6.78·9-s + 0.0286·10-s + 0.974·11-s − 6.22·12-s + 1.41·13-s − 0.877·15-s + 3.93·16-s − 6.28·17-s − 0.693·18-s − 1.00·19-s + 0.558·20-s − 0.0996·22-s + 1.39·23-s + 1.27·24-s − 4.92·25-s − 0.144·26-s + 11.8·27-s − 2.24·29-s + 0.0897·30-s + 9.08·31-s − 1.21·32-s + ⋯ |
| L(s) = 1 | − 0.0722·2-s + 1.80·3-s − 0.994·4-s − 0.125·5-s − 0.130·6-s + 0.144·8-s + 2.26·9-s + 0.00907·10-s + 0.293·11-s − 1.79·12-s + 0.392·13-s − 0.226·15-s + 0.984·16-s − 1.52·17-s − 0.163·18-s − 0.231·19-s + 0.124·20-s − 0.0212·22-s + 0.290·23-s + 0.260·24-s − 0.984·25-s − 0.0283·26-s + 2.27·27-s − 0.416·29-s + 0.0163·30-s + 1.63·31-s − 0.215·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6223 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6223 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.083674958\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.083674958\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 127 | \( 1 + T \) |
| good | 2 | \( 1 + 0.102T + 2T^{2} \) |
| 3 | \( 1 - 3.12T + 3T^{2} \) |
| 5 | \( 1 + 0.280T + 5T^{2} \) |
| 11 | \( 1 - 0.974T + 11T^{2} \) |
| 13 | \( 1 - 1.41T + 13T^{2} \) |
| 17 | \( 1 + 6.28T + 17T^{2} \) |
| 19 | \( 1 + 1.00T + 19T^{2} \) |
| 23 | \( 1 - 1.39T + 23T^{2} \) |
| 29 | \( 1 + 2.24T + 29T^{2} \) |
| 31 | \( 1 - 9.08T + 31T^{2} \) |
| 37 | \( 1 - 8.73T + 37T^{2} \) |
| 41 | \( 1 - 3.37T + 41T^{2} \) |
| 43 | \( 1 + 4.53T + 43T^{2} \) |
| 47 | \( 1 - 3.98T + 47T^{2} \) |
| 53 | \( 1 - 12.4T + 53T^{2} \) |
| 59 | \( 1 - 2.70T + 59T^{2} \) |
| 61 | \( 1 - 6.02T + 61T^{2} \) |
| 67 | \( 1 + 1.29T + 67T^{2} \) |
| 71 | \( 1 - 1.77T + 71T^{2} \) |
| 73 | \( 1 - 14.2T + 73T^{2} \) |
| 79 | \( 1 + 9.13T + 79T^{2} \) |
| 83 | \( 1 + 2.94T + 83T^{2} \) |
| 89 | \( 1 - 8.92T + 89T^{2} \) |
| 97 | \( 1 - 1.25T + 97T^{2} \) |
| 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
−8.341866559328969546725305190655, −7.62941603304052127952551365145, −6.86445158239349262347562281698, −6.00253907005092921634476158717, −4.81144698653989890571515900141, −4.12782302243824965941264084448, −3.79511679959731181346377644767, −2.76051167327498848042868348177, −2.05148859794030861897554170216, −0.874646779228700807523376516212,
0.874646779228700807523376516212, 2.05148859794030861897554170216, 2.76051167327498848042868348177, 3.79511679959731181346377644767, 4.12782302243824965941264084448, 4.81144698653989890571515900141, 6.00253907005092921634476158717, 6.86445158239349262347562281698, 7.62941603304052127952551365145, 8.341866559328969546725305190655
