| L(s) = 1 | − 10.3·3-s + 5·5-s − 7·7-s + 79.3·9-s + 41.8·11-s + 86.0·13-s − 51.5·15-s + 123.·17-s − 30.8·19-s + 72.1·21-s + 172.·23-s + 25·25-s − 539.·27-s − 73.6·29-s + 167.·31-s − 431.·33-s − 35·35-s − 5.02·37-s − 887.·39-s − 152.·41-s + 431.·43-s + 396.·45-s + 180.·47-s + 49·49-s − 1.26e3·51-s + 205.·53-s + 209.·55-s + ⋯ |
| L(s) = 1 | − 1.98·3-s + 0.447·5-s − 0.377·7-s + 2.93·9-s + 1.14·11-s + 1.83·13-s − 0.887·15-s + 1.75·17-s − 0.372·19-s + 0.750·21-s + 1.56·23-s + 0.200·25-s − 3.84·27-s − 0.471·29-s + 0.972·31-s − 2.27·33-s − 0.169·35-s − 0.0223·37-s − 3.64·39-s − 0.580·41-s + 1.53·43-s + 1.31·45-s + 0.559·47-s + 0.142·49-s − 3.48·51-s + 0.531·53-s + 0.513·55-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
| \(L(2)\) |
\(\approx\) |
\(1.631458432\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.631458432\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 10.3T + 27T^{2} \) |
| 11 | \( 1 - 41.8T + 1.33e3T^{2} \) |
| 13 | \( 1 - 86.0T + 2.19e3T^{2} \) |
| 17 | \( 1 - 123.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 30.8T + 6.85e3T^{2} \) |
| 23 | \( 1 - 172.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 73.6T + 2.43e4T^{2} \) |
| 31 | \( 1 - 167.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 5.02T + 5.06e4T^{2} \) |
| 41 | \( 1 + 152.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 431.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 180.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 205.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 91.2T + 2.05e5T^{2} \) |
| 61 | \( 1 + 531.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 291.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 575.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 91.0T + 3.89e5T^{2} \) |
| 79 | \( 1 - 113.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.09e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.03e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 869.T + 9.12e5T^{2} \) |
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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
−9.670462613506752668365888442372, −8.852594641578005196791326563081, −7.44701172871599235664616026310, −6.59690274469675564818368551071, −6.03289474575284030990056936712, −5.49500287449259547433309073146, −4.38002070410846075988037959288, −3.47982545416781974361446423376, −1.33657733298652250746620443017, −0.905081591657202615581037220077,
0.905081591657202615581037220077, 1.33657733298652250746620443017, 3.47982545416781974361446423376, 4.38002070410846075988037959288, 5.49500287449259547433309073146, 6.03289474575284030990056936712, 6.59690274469675564818368551071, 7.44701172871599235664616026310, 8.852594641578005196791326563081, 9.670462613506752668365888442372
