| L(s) = 1 | − 32·2-s + 1.02e3·4-s + 5.28e3·5-s − 4.90e4·7-s − 3.27e4·8-s − 1.68e5·10-s + 4.14e5·11-s − 5.22e5·13-s + 1.56e6·14-s + 1.04e6·16-s + 9.49e6·17-s + 1.30e7·19-s + 5.40e6·20-s − 1.32e7·22-s + 5.87e7·23-s − 2.09e7·25-s + 1.67e7·26-s − 5.02e7·28-s − 1.17e8·29-s + 1.42e8·31-s − 3.35e7·32-s − 3.03e8·34-s − 2.58e8·35-s + 7.18e8·37-s − 4.17e8·38-s − 1.73e8·40-s − 6.68e8·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.755·5-s − 1.10·7-s − 0.353·8-s − 0.534·10-s + 0.775·11-s − 0.390·13-s + 0.779·14-s + 1/4·16-s + 1.62·17-s + 1.20·19-s + 0.377·20-s − 0.548·22-s + 1.90·23-s − 0.429·25-s + 0.276·26-s − 0.551·28-s − 1.06·29-s + 0.896·31-s − 0.176·32-s − 1.14·34-s − 0.833·35-s + 1.70·37-s − 0.855·38-s − 0.267·40-s − 0.900·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(1.419538696\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.419538696\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p^{5} T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 1056 p T + p^{11} T^{2} \) |
| 7 | \( 1 + 49036 T + p^{11} T^{2} \) |
| 11 | \( 1 - 414336 T + p^{11} T^{2} \) |
| 13 | \( 1 + 522982 T + p^{11} T^{2} \) |
| 17 | \( 1 - 9499968 T + p^{11} T^{2} \) |
| 19 | \( 1 - 13053944 T + p^{11} T^{2} \) |
| 23 | \( 1 - 58755840 T + p^{11} T^{2} \) |
| 29 | \( 1 + 117142944 T + p^{11} T^{2} \) |
| 31 | \( 1 - 142907156 T + p^{11} T^{2} \) |
| 37 | \( 1 - 718521806 T + p^{11} T^{2} \) |
| 41 | \( 1 + 668055360 T + p^{11} T^{2} \) |
| 43 | \( 1 - 141575864 T + p^{11} T^{2} \) |
| 47 | \( 1 - 729235200 T + p^{11} T^{2} \) |
| 53 | \( 1 - 4917225312 T + p^{11} T^{2} \) |
| 59 | \( 1 - 1408015104 T + p^{11} T^{2} \) |
| 61 | \( 1 + 3223327018 T + p^{11} T^{2} \) |
| 67 | \( 1 + 2358681328 T + p^{11} T^{2} \) |
| 71 | \( 1 + 22245092352 T + p^{11} T^{2} \) |
| 73 | \( 1 + 28036594330 T + p^{11} T^{2} \) |
| 79 | \( 1 + 20685045676 T + p^{11} T^{2} \) |
| 83 | \( 1 - 37818604416 T + p^{11} T^{2} \) |
| 89 | \( 1 - 11288711808 T + p^{11} T^{2} \) |
| 97 | \( 1 + 115724393266 T + p^{11} T^{2} \) |
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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
−16.41645533351426132334190145777, −14.80983040441961086419425228250, −13.29497597807711910275053434514, −11.83784215206999943295772583103, −10.00878332351374057165802516783, −9.281314900169264685882112156273, −7.26221865194666204152221621769, −5.79868717158306993480903337012, −3.07276826320355316617926546304, −1.05540292367277643148934224106,
1.05540292367277643148934224106, 3.07276826320355316617926546304, 5.79868717158306993480903337012, 7.26221865194666204152221621769, 9.281314900169264685882112156273, 10.00878332351374057165802516783, 11.83784215206999943295772583103, 13.29497597807711910275053434514, 14.80983040441961086419425228250, 16.41645533351426132334190145777
