| L(s) = 1 | − 3·3-s + 7.31·5-s + 27.4·7-s + 9·9-s + 39.7·11-s + 57.5·13-s − 21.9·15-s − 10.2·17-s − 52.7·19-s − 82.3·21-s + 23·23-s − 71.5·25-s − 27·27-s + 126.·29-s − 146.·31-s − 119.·33-s + 200.·35-s − 90.3·37-s − 172.·39-s + 61.1·41-s + 497.·43-s + 65.8·45-s + 102.·47-s + 410.·49-s + 30.7·51-s − 721.·53-s + 291.·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.654·5-s + 1.48·7-s + 0.333·9-s + 1.09·11-s + 1.22·13-s − 0.377·15-s − 0.146·17-s − 0.636·19-s − 0.855·21-s + 0.208·23-s − 0.572·25-s − 0.192·27-s + 0.807·29-s − 0.847·31-s − 0.629·33-s + 0.969·35-s − 0.401·37-s − 0.709·39-s + 0.232·41-s + 1.76·43-s + 0.218·45-s + 0.319·47-s + 1.19·49-s + 0.0845·51-s − 1.86·53-s + 0.713·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.512028732\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.512028732\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 3T \) |
| 23 | \( 1 - 23T \) |
| good | 5 | \( 1 - 7.31T + 125T^{2} \) |
| 7 | \( 1 - 27.4T + 343T^{2} \) |
| 11 | \( 1 - 39.7T + 1.33e3T^{2} \) |
| 13 | \( 1 - 57.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 10.2T + 4.91e3T^{2} \) |
| 19 | \( 1 + 52.7T + 6.85e3T^{2} \) |
| 29 | \( 1 - 126.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 146.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 90.3T + 5.06e4T^{2} \) |
| 41 | \( 1 - 61.1T + 6.89e4T^{2} \) |
| 43 | \( 1 - 497.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 102.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 721.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 234.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 507.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 58.4T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.15e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 660.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.39e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 81.5T + 5.71e5T^{2} \) |
| 89 | \( 1 + 185.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 830.T + 9.12e5T^{2} \) |
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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
−10.72333239338764522635719584312, −9.430540420609958118230234555437, −8.676097764771118105162416112154, −7.72944715710183126585735769360, −6.51673604665231398389287440982, −5.83756694330983255693371669468, −4.78238024789666844304190774673, −3.86059166976110884683220124098, −1.96762990670601376326323064255, −1.11257299332890150261943493110,
1.11257299332890150261943493110, 1.96762990670601376326323064255, 3.86059166976110884683220124098, 4.78238024789666844304190774673, 5.83756694330983255693371669468, 6.51673604665231398389287440982, 7.72944715710183126585735769360, 8.676097764771118105162416112154, 9.430540420609958118230234555437, 10.72333239338764522635719584312
