| L(s) = 1 | + 1.45e3·3-s + 4.09e3·4-s + 7.96e4·7-s + 1.57e6·9-s + 2.79e5·11-s + 5.94e6·12-s + 1.67e7·16-s − 1.70e7·17-s + 1.15e8·21-s − 2.31e8·23-s + 2.44e8·25-s + 1.52e9·27-s + 3.26e8·28-s − 1.06e9·29-s + 1.58e9·31-s + 4.05e8·33-s + 6.46e9·36-s + 3.45e9·37-s − 9.47e9·41-s + 1.14e9·44-s + 2.43e10·48-s − 7.49e9·49-s − 2.46e10·51-s − 3.53e10·59-s − 9.17e10·61-s + 1.25e11·63-s + 6.87e10·64-s + ⋯ |
| L(s) = 1 | + 1.99·3-s + 4-s + 0.677·7-s + 2.97·9-s + 0.157·11-s + 1.99·12-s + 16-s − 0.704·17-s + 1.34·21-s − 1.56·23-s + 25-s + 3.92·27-s + 0.677·28-s − 1.79·29-s + 1.78·31-s + 0.314·33-s + 2.97·36-s + 1.34·37-s − 1.99·41-s + 0.157·44-s + 1.99·48-s − 0.541·49-s − 1.40·51-s − 0.837·59-s − 1.78·61-s + 2.01·63-s + 64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{13}{2})\) |
\(\approx\) |
\(6.886385444\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.886385444\) |
| \(L(7)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 83 | \( 1 - 3.26e11T \) |
| good | 2 | \( 1 - 4.09e3T^{2} \) |
| 3 | \( 1 - 1.45e3T + 5.31e5T^{2} \) |
| 5 | \( 1 - 2.44e8T^{2} \) |
| 7 | \( 1 - 7.96e4T + 1.38e10T^{2} \) |
| 11 | \( 1 - 2.79e5T + 3.13e12T^{2} \) |
| 13 | \( 1 - 2.32e13T^{2} \) |
| 17 | \( 1 + 1.70e7T + 5.82e14T^{2} \) |
| 19 | \( 1 - 2.21e15T^{2} \) |
| 23 | \( 1 + 2.31e8T + 2.19e16T^{2} \) |
| 29 | \( 1 + 1.06e9T + 3.53e17T^{2} \) |
| 31 | \( 1 - 1.58e9T + 7.87e17T^{2} \) |
| 37 | \( 1 - 3.45e9T + 6.58e18T^{2} \) |
| 41 | \( 1 + 9.47e9T + 2.25e19T^{2} \) |
| 43 | \( 1 - 3.99e19T^{2} \) |
| 47 | \( 1 - 1.16e20T^{2} \) |
| 53 | \( 1 - 4.91e20T^{2} \) |
| 59 | \( 1 + 3.53e10T + 1.77e21T^{2} \) |
| 61 | \( 1 + 9.17e10T + 2.65e21T^{2} \) |
| 67 | \( 1 - 8.18e21T^{2} \) |
| 71 | \( 1 - 1.64e22T^{2} \) |
| 73 | \( 1 - 2.29e22T^{2} \) |
| 79 | \( 1 - 5.90e22T^{2} \) |
| 89 | \( 1 - 2.46e23T^{2} \) |
| 97 | \( 1 - 6.93e23T^{2} \) |
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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
−11.94077177002931132442095274994, −10.58690422575881913628502002089, −9.518712470686704764474983968535, −8.321346646336957397823816654304, −7.67908052544686140765495496440, −6.54739404053945269896670506587, −4.45047896146680919095433953157, −3.24252135428939675486723024825, −2.21511776253716780369490012603, −1.46964608485394452661783844332,
1.46964608485394452661783844332, 2.21511776253716780369490012603, 3.24252135428939675486723024825, 4.45047896146680919095433953157, 6.54739404053945269896670506587, 7.67908052544686140765495496440, 8.321346646336957397823816654304, 9.518712470686704764474983968535, 10.58690422575881913628502002089, 11.94077177002931132442095274994
