| L(s) = 1 | − 6.38·3-s + 5·5-s − 7·7-s + 13.7·9-s − 55.5·11-s − 46.0·13-s − 31.9·15-s + 11.5·17-s − 89.4·19-s + 44.6·21-s − 63.2·23-s + 25·25-s + 84.7·27-s + 4.22·29-s + 19.0·31-s + 354.·33-s − 35·35-s + 30.6·37-s + 293.·39-s − 33.7·41-s − 363.·43-s + 68.6·45-s − 439.·47-s + 49·49-s − 73.6·51-s − 295.·53-s − 277.·55-s + ⋯ |
| L(s) = 1 | − 1.22·3-s + 0.447·5-s − 0.377·7-s + 0.508·9-s − 1.52·11-s − 0.981·13-s − 0.549·15-s + 0.164·17-s − 1.08·19-s + 0.464·21-s − 0.573·23-s + 0.200·25-s + 0.603·27-s + 0.0270·29-s + 0.110·31-s + 1.87·33-s − 0.169·35-s + 0.136·37-s + 1.20·39-s − 0.128·41-s − 1.28·43-s + 0.227·45-s − 1.36·47-s + 0.142·49-s − 0.202·51-s − 0.766·53-s − 0.681·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.3916841542\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3916841542\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| 7 | \( 1 + 7T \) |
| good | 3 | \( 1 + 6.38T + 27T^{2} \) |
| 11 | \( 1 + 55.5T + 1.33e3T^{2} \) |
| 13 | \( 1 + 46.0T + 2.19e3T^{2} \) |
| 17 | \( 1 - 11.5T + 4.91e3T^{2} \) |
| 19 | \( 1 + 89.4T + 6.85e3T^{2} \) |
| 23 | \( 1 + 63.2T + 1.21e4T^{2} \) |
| 29 | \( 1 - 4.22T + 2.43e4T^{2} \) |
| 31 | \( 1 - 19.0T + 2.97e4T^{2} \) |
| 37 | \( 1 - 30.6T + 5.06e4T^{2} \) |
| 41 | \( 1 + 33.7T + 6.89e4T^{2} \) |
| 43 | \( 1 + 363.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 439.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 295.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 92.0T + 2.05e5T^{2} \) |
| 61 | \( 1 + 230.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 20.2T + 3.00e5T^{2} \) |
| 71 | \( 1 - 284.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 333.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 618.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 490.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 108.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.65e3T + 9.12e5T^{2} \) |
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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
−9.886107749104124467677001829349, −8.587361626784405088126863092319, −7.72169336350001051779747879716, −6.71652445586503827079047901620, −6.02805961543643779839051478369, −5.21429579752279234422078867854, −4.62742161661247971534063171378, −3.05557826671755710515279449736, −2.01141708924242008097220223330, −0.32584377358104612954104689038,
0.32584377358104612954104689038, 2.01141708924242008097220223330, 3.05557826671755710515279449736, 4.62742161661247971534063171378, 5.21429579752279234422078867854, 6.02805961543643779839051478369, 6.71652445586503827079047901620, 7.72169336350001051779747879716, 8.587361626784405088126863092319, 9.886107749104124467677001829349
