| L(s) = 1 | + 32·2-s + 243·3-s + 1.02e3·4-s − 1.18e4·5-s + 7.77e3·6-s + 1.68e4·7-s + 3.27e4·8-s + 5.90e4·9-s − 3.80e5·10-s + 7.27e5·11-s + 2.48e5·12-s − 1.89e6·13-s + 5.37e5·14-s − 2.88e6·15-s + 1.04e6·16-s − 1.02e7·17-s + 1.88e6·18-s − 1.27e7·19-s − 1.21e7·20-s + 4.08e6·21-s + 2.32e7·22-s − 2.74e7·23-s + 7.96e6·24-s + 9.23e7·25-s − 6.06e7·26-s + 1.43e7·27-s + 1.72e7·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 1.70·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s − 1.20·10-s + 1.36·11-s + 0.288·12-s − 1.41·13-s + 0.267·14-s − 0.981·15-s + 1/4·16-s − 1.74·17-s + 0.235·18-s − 1.18·19-s − 0.850·20-s + 0.218·21-s + 0.962·22-s − 0.888·23-s + 0.204·24-s + 1.89·25-s − 1.00·26-s + 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{13}{2})\) |
|
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 | \( 1 - p^{5} T \) |
| 7 | \( 1 - p^{5} T \) |
| good | 5 | \( 1 + 2376 p T + p^{11} T^{2} \) |
| 11 | \( 1 - 727110 T + p^{11} T^{2} \) |
| 13 | \( 1 + 1895734 T + p^{11} T^{2} \) |
| 17 | \( 1 + 10233912 T + p^{11} T^{2} \) |
| 19 | \( 1 + 12792796 T + p^{11} T^{2} \) |
| 23 | \( 1 + 27412290 T + p^{11} T^{2} \) |
| 29 | \( 1 + 108082914 T + p^{11} T^{2} \) |
| 31 | \( 1 + 202243384 T + p^{11} T^{2} \) |
| 37 | \( 1 - 412454954 T + p^{11} T^{2} \) |
| 41 | \( 1 + 245108604 T + p^{11} T^{2} \) |
| 43 | \( 1 - 509839844 T + p^{11} T^{2} \) |
| 47 | \( 1 - 699876996 T + p^{11} T^{2} \) |
| 53 | \( 1 + 4836690462 T + p^{11} T^{2} \) |
| 59 | \( 1 - 2462248272 T + p^{11} T^{2} \) |
| 61 | \( 1 - 9054716702 T + p^{11} T^{2} \) |
| 67 | \( 1 + 2923148584 T + p^{11} T^{2} \) |
| 71 | \( 1 + 11268869310 T + p^{11} T^{2} \) |
| 73 | \( 1 - 2809231382 T + p^{11} T^{2} \) |
| 79 | \( 1 - 25573574516 T + p^{11} T^{2} \) |
| 83 | \( 1 + 38718767364 T + p^{11} T^{2} \) |
| 89 | \( 1 - 90783707844 T + p^{11} T^{2} \) |
| 97 | \( 1 - 130559880038 T + p^{11} T^{2} \) |
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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
−12.89105944608494566863606099144, −11.89420202794563870934809013727, −11.03381367252348981336070874982, −9.060028542786688813559978259033, −7.79395657700024210084483133987, −6.77294364027559558614336647064, −4.48380901071728712430225115209, −3.86526905058444319357505602261, −2.14694460629424843299506192267, 0,
2.14694460629424843299506192267, 3.86526905058444319357505602261, 4.48380901071728712430225115209, 6.77294364027559558614336647064, 7.79395657700024210084483133987, 9.060028542786688813559978259033, 11.03381367252348981336070874982, 11.89420202794563870934809013727, 12.89105944608494566863606099144
