| L(s) = 1 | + 9.65·3-s − 3.08·5-s − 7·7-s + 66.1·9-s − 11·11-s + 13.5·13-s − 29.8·15-s − 87.3·17-s + 23.4·19-s − 67.5·21-s + 193.·23-s − 115.·25-s + 378.·27-s + 224.·29-s + 84.0·31-s − 106.·33-s + 21.6·35-s + 138.·37-s + 130.·39-s + 404.·41-s + 138.·43-s − 204.·45-s + 198.·47-s + 49·49-s − 843.·51-s + 423.·53-s + 33.9·55-s + ⋯ |
| L(s) = 1 | + 1.85·3-s − 0.276·5-s − 0.377·7-s + 2.45·9-s − 0.301·11-s + 0.288·13-s − 0.513·15-s − 1.24·17-s + 0.283·19-s − 0.702·21-s + 1.75·23-s − 0.923·25-s + 2.69·27-s + 1.43·29-s + 0.486·31-s − 0.560·33-s + 0.104·35-s + 0.615·37-s + 0.535·39-s + 1.54·41-s + 0.489·43-s − 0.676·45-s + 0.614·47-s + 0.142·49-s − 2.31·51-s + 1.09·53-s + 0.0832·55-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
| \(L(2)\) |
\(\approx\) |
\(4.265290730\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.265290730\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 9.65T + 27T^{2} \) |
| 5 | \( 1 + 3.08T + 125T^{2} \) |
| 13 | \( 1 - 13.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 87.3T + 4.91e3T^{2} \) |
| 19 | \( 1 - 23.4T + 6.85e3T^{2} \) |
| 23 | \( 1 - 193.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 224.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 84.0T + 2.97e4T^{2} \) |
| 37 | \( 1 - 138.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 404.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 138.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 198.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 423.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 311.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 262.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 441.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 731.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 723.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 647.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 72.9T + 5.71e5T^{2} \) |
| 89 | \( 1 + 243.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.61e3T + 9.12e5T^{2} \) |
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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
−8.938740440866195368360131430843, −8.829314113811933063775135387486, −7.73850448872567094944555260114, −7.21534787953682923233056128026, −6.23184136553215791047383981832, −4.69162685903982842033479751080, −3.97899842052922618543110053169, −2.95233578187767413045587384577, −2.39658078810104901329302519693, −0.998819271263063248124085349073,
0.998819271263063248124085349073, 2.39658078810104901329302519693, 2.95233578187767413045587384577, 3.97899842052922618543110053169, 4.69162685903982842033479751080, 6.23184136553215791047383981832, 7.21534787953682923233056128026, 7.73850448872567094944555260114, 8.829314113811933063775135387486, 8.938740440866195368360131430843
