| L(s) = 1 | + 1.97e3·5-s − 5.58e4·7-s − 8.38e5·11-s − 6.10e5·13-s − 2.80e6·17-s − 1.12e7·19-s + 1.99e7·23-s − 4.49e7·25-s + 1.15e8·29-s − 1.51e7·31-s − 1.10e8·35-s + 2.30e8·37-s − 3.56e8·41-s + 7.19e8·43-s − 7.29e8·47-s + 1.13e9·49-s + 5.76e9·53-s − 1.65e9·55-s − 6.44e9·59-s − 8.76e9·61-s − 1.20e9·65-s − 1.88e9·67-s − 2.25e10·71-s + 2.47e10·73-s + 4.67e10·77-s + 2.12e10·79-s − 3.57e10·83-s + ⋯ |
| L(s) = 1 | + 0.282·5-s − 1.25·7-s − 1.56·11-s − 0.456·13-s − 0.479·17-s − 1.04·19-s + 0.646·23-s − 0.920·25-s + 1.04·29-s − 0.0949·31-s − 0.354·35-s + 0.546·37-s − 0.480·41-s + 0.746·43-s − 0.464·47-s + 0.574·49-s + 1.89·53-s − 0.443·55-s − 1.17·59-s − 1.32·61-s − 0.128·65-s − 0.170·67-s − 1.48·71-s + 1.39·73-s + 1.96·77-s + 0.775·79-s − 0.996·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(0.8472769448\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8472769448\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 1.97e3T + 4.88e7T^{2} \) |
| 7 | \( 1 + 5.58e4T + 1.97e9T^{2} \) |
| 11 | \( 1 + 8.38e5T + 2.85e11T^{2} \) |
| 13 | \( 1 + 6.10e5T + 1.79e12T^{2} \) |
| 17 | \( 1 + 2.80e6T + 3.42e13T^{2} \) |
| 19 | \( 1 + 1.12e7T + 1.16e14T^{2} \) |
| 23 | \( 1 - 1.99e7T + 9.52e14T^{2} \) |
| 29 | \( 1 - 1.15e8T + 1.22e16T^{2} \) |
| 31 | \( 1 + 1.51e7T + 2.54e16T^{2} \) |
| 37 | \( 1 - 2.30e8T + 1.77e17T^{2} \) |
| 41 | \( 1 + 3.56e8T + 5.50e17T^{2} \) |
| 43 | \( 1 - 7.19e8T + 9.29e17T^{2} \) |
| 47 | \( 1 + 7.29e8T + 2.47e18T^{2} \) |
| 53 | \( 1 - 5.76e9T + 9.26e18T^{2} \) |
| 59 | \( 1 + 6.44e9T + 3.01e19T^{2} \) |
| 61 | \( 1 + 8.76e9T + 4.35e19T^{2} \) |
| 67 | \( 1 + 1.88e9T + 1.22e20T^{2} \) |
| 71 | \( 1 + 2.25e10T + 2.31e20T^{2} \) |
| 73 | \( 1 - 2.47e10T + 3.13e20T^{2} \) |
| 79 | \( 1 - 2.12e10T + 7.47e20T^{2} \) |
| 83 | \( 1 + 3.57e10T + 1.28e21T^{2} \) |
| 89 | \( 1 - 9.42e10T + 2.77e21T^{2} \) |
| 97 | \( 1 - 8.97e10T + 7.15e21T^{2} \) |
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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
−10.74474179578264720126245836141, −10.07734550594652545229265946075, −9.071459556191600628983676870443, −7.86941637569196484869740422791, −6.72382717136108129106653128301, −5.75719277695783484956957554349, −4.53723898547811333443160831153, −3.06989092340088711663240940577, −2.22597533965597931333644672319, −0.40610027033039708598518447261,
0.40610027033039708598518447261, 2.22597533965597931333644672319, 3.06989092340088711663240940577, 4.53723898547811333443160831153, 5.75719277695783484956957554349, 6.72382717136108129106653128301, 7.86941637569196484869740422791, 9.071459556191600628983676870443, 10.07734550594652545229265946075, 10.74474179578264720126245836141
