| L(s) = 1 | + 36·3-s + 3.49e3·5-s − 5.54e4·7-s − 1.75e5·9-s + 5.97e5·11-s − 1.37e6·13-s + 1.25e5·15-s + 1.01e7·17-s + 7.29e6·19-s − 1.99e6·21-s − 3.20e7·23-s − 3.66e7·25-s − 1.27e7·27-s + 1.36e7·29-s + 2.33e8·31-s + 2.14e7·33-s − 1.93e8·35-s + 2.57e8·37-s − 4.94e7·39-s − 2.21e8·41-s + 1.69e9·43-s − 6.13e8·45-s + 5.27e8·47-s + 1.09e9·49-s + 3.65e8·51-s − 3.27e9·53-s + 2.08e9·55-s + ⋯ |
| L(s) = 1 | + 0.0855·3-s + 0.499·5-s − 1.24·7-s − 0.992·9-s + 1.11·11-s − 1.02·13-s + 0.0427·15-s + 1.73·17-s + 0.676·19-s − 0.106·21-s − 1.03·23-s − 0.750·25-s − 0.170·27-s + 0.123·29-s + 1.46·31-s + 0.0955·33-s − 0.622·35-s + 0.611·37-s − 0.0877·39-s − 0.298·41-s + 1.76·43-s − 0.495·45-s + 0.335·47-s + 0.555·49-s + 0.148·51-s − 1.07·53-s + 0.558·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(1.827066442\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.827066442\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 - 4 p^{2} T + p^{11} T^{2} \) |
| 5 | \( 1 - 698 p T + p^{11} T^{2} \) |
| 7 | \( 1 + 55464 T + p^{11} T^{2} \) |
| 11 | \( 1 - 597004 T + p^{11} T^{2} \) |
| 13 | \( 1 + 1373878 T + p^{11} T^{2} \) |
| 17 | \( 1 - 10140850 T + p^{11} T^{2} \) |
| 19 | \( 1 - 7297396 T + p^{11} T^{2} \) |
| 23 | \( 1 + 32057464 T + p^{11} T^{2} \) |
| 29 | \( 1 - 13605402 T + p^{11} T^{2} \) |
| 31 | \( 1 - 233160800 T + p^{11} T^{2} \) |
| 37 | \( 1 - 6967194 p T + p^{11} T^{2} \) |
| 41 | \( 1 + 221438598 T + p^{11} T^{2} \) |
| 43 | \( 1 - 1697758892 T + p^{11} T^{2} \) |
| 47 | \( 1 - 527509392 T + p^{11} T^{2} \) |
| 53 | \( 1 + 3277379822 T + p^{11} T^{2} \) |
| 59 | \( 1 - 3001908988 T + p^{11} T^{2} \) |
| 61 | \( 1 - 11630023610 T + p^{11} T^{2} \) |
| 67 | \( 1 - 17189000548 T + p^{11} T^{2} \) |
| 71 | \( 1 - 26169539608 T + p^{11} T^{2} \) |
| 73 | \( 1 + 7039021094 T + p^{11} T^{2} \) |
| 79 | \( 1 + 4199910416 T + p^{11} T^{2} \) |
| 83 | \( 1 - 39739936436 T + p^{11} T^{2} \) |
| 89 | \( 1 - 10565331594 T + p^{11} T^{2} \) |
| 97 | \( 1 + 69851645662 T + p^{11} T^{2} \) |
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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
−12.45668382473351312615554333949, −11.72584783519023225291469795971, −9.905685978581069379140422685032, −9.508721943947301193099613619122, −7.926690729851102073366879308693, −6.46611729087608056703385538911, −5.55290944297178723041532672182, −3.66975868734989645658656334788, −2.53084297418135346805085339967, −0.75007900099107823663901473268,
0.75007900099107823663901473268, 2.53084297418135346805085339967, 3.66975868734989645658656334788, 5.55290944297178723041532672182, 6.46611729087608056703385538911, 7.926690729851102073366879308693, 9.508721943947301193099613619122, 9.905685978581069379140422685032, 11.72584783519023225291469795971, 12.45668382473351312615554333949
