| L(s) = 1 | − 4.49·3-s − 8.18·5-s − 7·7-s − 6.81·9-s − 11·11-s + 6.07·13-s + 36.7·15-s − 108.·17-s − 40.0·19-s + 31.4·21-s − 49.1·23-s − 58.0·25-s + 151.·27-s − 89.1·29-s − 111.·31-s + 49.4·33-s + 57.2·35-s − 195.·37-s − 27.2·39-s − 226.·41-s + 17.0·43-s + 55.7·45-s − 61.5·47-s + 49·49-s + 487.·51-s − 691.·53-s + 90.0·55-s + ⋯ |
| L(s) = 1 | − 0.864·3-s − 0.731·5-s − 0.377·7-s − 0.252·9-s − 0.301·11-s + 0.129·13-s + 0.632·15-s − 1.54·17-s − 0.483·19-s + 0.326·21-s − 0.445·23-s − 0.464·25-s + 1.08·27-s − 0.570·29-s − 0.646·31-s + 0.260·33-s + 0.276·35-s − 0.867·37-s − 0.111·39-s − 0.864·41-s + 0.0604·43-s + 0.184·45-s − 0.190·47-s + 0.142·49-s + 1.33·51-s − 1.79·53-s + 0.220·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.1424219751\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1424219751\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + 7T \) |
| 11 | \( 1 + 11T \) |
| good | 3 | \( 1 + 4.49T + 27T^{2} \) |
| 5 | \( 1 + 8.18T + 125T^{2} \) |
| 13 | \( 1 - 6.07T + 2.19e3T^{2} \) |
| 17 | \( 1 + 108.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 40.0T + 6.85e3T^{2} \) |
| 23 | \( 1 + 49.1T + 1.21e4T^{2} \) |
| 29 | \( 1 + 89.1T + 2.43e4T^{2} \) |
| 31 | \( 1 + 111.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 195.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 226.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 17.0T + 7.95e4T^{2} \) |
| 47 | \( 1 + 61.5T + 1.03e5T^{2} \) |
| 53 | \( 1 + 691.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 240.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 694.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 396.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 232.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 43.3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 230.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 210.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 328.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.76e3T + 9.12e5T^{2} \) |
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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
−9.273027516232030265382959712178, −8.536914283694565571575533201105, −7.66806039132349485230454792622, −6.70223096581878018545019723737, −6.08518604768821151966080712377, −5.10493040617229702106225215348, −4.24578897107977906935667521682, −3.24954139827315818334746568158, −1.94476638021518614291412281208, −0.18991349167247393819029701601,
0.18991349167247393819029701601, 1.94476638021518614291412281208, 3.24954139827315818334746568158, 4.24578897107977906935667521682, 5.10493040617229702106225215348, 6.08518604768821151966080712377, 6.70223096581878018545019723737, 7.66806039132349485230454792622, 8.536914283694565571575533201105, 9.273027516232030265382959712178
