| L(s) = 1 | + 6.24·3-s + 18.9·5-s + 7·7-s + 12.0·9-s + 49.7·11-s + 38.6·13-s + 118.·15-s − 83.9·17-s − 136.·19-s + 43.7·21-s + 149.·23-s + 233.·25-s − 93.4·27-s − 63.8·29-s + 6.03·31-s + 310.·33-s + 132.·35-s − 37.1·37-s + 241.·39-s − 361.·41-s + 99.2·43-s + 227.·45-s − 343.·47-s + 49·49-s − 524.·51-s − 625.·53-s + 941.·55-s + ⋯ |
| L(s) = 1 | + 1.20·3-s + 1.69·5-s + 0.377·7-s + 0.445·9-s + 1.36·11-s + 0.825·13-s + 2.03·15-s − 1.19·17-s − 1.65·19-s + 0.454·21-s + 1.35·23-s + 1.86·25-s − 0.666·27-s − 0.408·29-s + 0.0349·31-s + 1.63·33-s + 0.639·35-s − 0.165·37-s + 0.992·39-s − 1.37·41-s + 0.351·43-s + 0.754·45-s − 1.06·47-s + 0.142·49-s − 1.44·51-s − 1.62·53-s + 2.30·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.260778579\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.260778579\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - 7T \) |
| good | 3 | \( 1 - 6.24T + 27T^{2} \) |
| 5 | \( 1 - 18.9T + 125T^{2} \) |
| 11 | \( 1 - 49.7T + 1.33e3T^{2} \) |
| 13 | \( 1 - 38.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + 83.9T + 4.91e3T^{2} \) |
| 19 | \( 1 + 136.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 149.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 63.8T + 2.43e4T^{2} \) |
| 31 | \( 1 - 6.03T + 2.97e4T^{2} \) |
| 37 | \( 1 + 37.1T + 5.06e4T^{2} \) |
| 41 | \( 1 + 361.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 99.2T + 7.95e4T^{2} \) |
| 47 | \( 1 + 343.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 625.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 363.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 120.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 173.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 468.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 710.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.07e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.37e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 462.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 456.T + 9.12e5T^{2} \) |
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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
−10.58523452017150786063502151022, −9.403183774577296654849964494237, −8.980500399486457721998049089435, −8.362690758494077919345740064777, −6.74013992672965828007942799618, −6.22243881290069564244889619245, −4.82748988150644923089449238412, −3.56860009610892970221911470996, −2.25943544751598209828646700376, −1.54950844054971783623350626654,
1.54950844054971783623350626654, 2.25943544751598209828646700376, 3.56860009610892970221911470996, 4.82748988150644923089449238412, 6.22243881290069564244889619245, 6.74013992672965828007942799618, 8.362690758494077919345740064777, 8.980500399486457721998049089435, 9.403183774577296654849964494237, 10.58523452017150786063502151022
