| L(s) = 1 | − 4.88·2-s − 5.33·3-s + 15.9·4-s − 4.56·5-s + 26.0·6-s + 7·7-s − 38.6·8-s + 1.40·9-s + 22.3·10-s + 12.2·11-s − 84.7·12-s − 2.40·13-s − 34.2·14-s + 24.3·15-s + 61.6·16-s − 6.89·18-s + 81.2·19-s − 72.6·20-s − 37.3·21-s − 59.7·22-s + 67.0·23-s + 205.·24-s − 104.·25-s + 11.7·26-s + 136.·27-s + 111.·28-s + 54.9·29-s + ⋯ |
| L(s) = 1 | − 1.72·2-s − 1.02·3-s + 1.98·4-s − 0.408·5-s + 1.77·6-s + 0.377·7-s − 1.70·8-s + 0.0522·9-s + 0.706·10-s + 0.335·11-s − 2.03·12-s − 0.0513·13-s − 0.653·14-s + 0.419·15-s + 0.963·16-s − 0.0902·18-s + 0.980·19-s − 0.812·20-s − 0.387·21-s − 0.579·22-s + 0.607·23-s + 1.75·24-s − 0.832·25-s + 0.0887·26-s + 0.972·27-s + 0.751·28-s + 0.351·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2023 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2023 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.4624912025\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4624912025\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - 7T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 + 4.88T + 8T^{2} \) |
| 3 | \( 1 + 5.33T + 27T^{2} \) |
| 5 | \( 1 + 4.56T + 125T^{2} \) |
| 11 | \( 1 - 12.2T + 1.33e3T^{2} \) |
| 13 | \( 1 + 2.40T + 2.19e3T^{2} \) |
| 19 | \( 1 - 81.2T + 6.85e3T^{2} \) |
| 23 | \( 1 - 67.0T + 1.21e4T^{2} \) |
| 29 | \( 1 - 54.9T + 2.43e4T^{2} \) |
| 31 | \( 1 + 206.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 83.7T + 5.06e4T^{2} \) |
| 41 | \( 1 + 30.4T + 6.89e4T^{2} \) |
| 43 | \( 1 - 256.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 20.2T + 1.03e5T^{2} \) |
| 53 | \( 1 - 17.1T + 1.48e5T^{2} \) |
| 59 | \( 1 - 693.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 278.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 893.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 180.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 273.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.06e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 984.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 633.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 233.T + 9.12e5T^{2} \) |
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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
−8.806923666417010097051333705531, −8.100097553944573999702476480465, −7.32535106099066703734257944057, −6.76239331297753183508415150560, −5.79455296005901131838459076495, −5.03986609584811586108698110161, −3.73701627608776268668384436527, −2.43790713676802590410538396843, −1.26946703191567540408108481370, −0.48719803552469556980480315514,
0.48719803552469556980480315514, 1.26946703191567540408108481370, 2.43790713676802590410538396843, 3.73701627608776268668384436527, 5.03986609584811586108698110161, 5.79455296005901131838459076495, 6.76239331297753183508415150560, 7.32535106099066703734257944057, 8.100097553944573999702476480465, 8.806923666417010097051333705531
