| L(s) = 1 | + 2.62·2-s + 0.447·3-s − 1.10·4-s + 9.81·5-s + 1.17·6-s − 21.1·7-s − 23.9·8-s − 26.7·9-s + 25.7·10-s − 0.492·12-s + 31.4·13-s − 55.5·14-s + 4.39·15-s − 53.9·16-s + 17·17-s − 70.3·18-s − 164.·19-s − 10.8·20-s − 9.46·21-s + 118.·23-s − 10.6·24-s − 28.6·25-s + 82.6·26-s − 24.0·27-s + 23.2·28-s − 40.9·29-s + 11.5·30-s + ⋯ |
| L(s) = 1 | + 0.928·2-s + 0.0861·3-s − 0.137·4-s + 0.877·5-s + 0.0799·6-s − 1.14·7-s − 1.05·8-s − 0.992·9-s + 0.815·10-s − 0.0118·12-s + 0.671·13-s − 1.06·14-s + 0.0756·15-s − 0.843·16-s + 0.242·17-s − 0.921·18-s − 1.98·19-s − 0.120·20-s − 0.0983·21-s + 1.07·23-s − 0.0909·24-s − 0.229·25-s + 0.623·26-s − 0.171·27-s + 0.157·28-s − 0.262·29-s + 0.0702·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2057 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2057 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.198523133\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.198523133\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 \) |
| 17 | \( 1 - 17T \) |
| good | 2 | \( 1 - 2.62T + 8T^{2} \) |
| 3 | \( 1 - 0.447T + 27T^{2} \) |
| 5 | \( 1 - 9.81T + 125T^{2} \) |
| 7 | \( 1 + 21.1T + 343T^{2} \) |
| 13 | \( 1 - 31.4T + 2.19e3T^{2} \) |
| 19 | \( 1 + 164.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 118.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 40.9T + 2.43e4T^{2} \) |
| 31 | \( 1 - 229.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 196.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 328.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 34.5T + 7.95e4T^{2} \) |
| 47 | \( 1 - 281.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 89.1T + 1.48e5T^{2} \) |
| 59 | \( 1 - 218.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 113.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 38.6T + 3.00e5T^{2} \) |
| 71 | \( 1 + 242.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 345.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 959.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 109.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.34e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.24e3T + 9.12e5T^{2} \) |
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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
−8.850739169252968343424992140465, −8.228666655381297105636382466556, −6.68954815593001923645217959825, −6.20573738771098753455980974632, −5.71700600090009514483017735933, −4.76177059375676213247976247314, −3.78814517587150982647091586801, −3.01959334615633107969218910864, −2.24615791773116403249327905249, −0.55783514327573564066075491942,
0.55783514327573564066075491942, 2.24615791773116403249327905249, 3.01959334615633107969218910864, 3.78814517587150982647091586801, 4.76177059375676213247976247314, 5.71700600090009514483017735933, 6.20573738771098753455980974632, 6.68954815593001923645217959825, 8.228666655381297105636382466556, 8.850739169252968343424992140465
