| L(s) = 1 | + 3-s + 5-s − 7-s + 9-s − 5·11-s − 13-s + 15-s − 17-s + 2·19-s − 21-s − 6·23-s − 4·25-s + 27-s − 2·29-s + 8·31-s − 5·33-s − 35-s − 37-s − 39-s + 8·41-s + 43-s + 45-s + 4·47-s + 49-s − 51-s + 5·53-s − 5·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 0.377·7-s + 1/3·9-s − 1.50·11-s − 0.277·13-s + 0.258·15-s − 0.242·17-s + 0.458·19-s − 0.218·21-s − 1.25·23-s − 4/5·25-s + 0.192·27-s − 0.371·29-s + 1.43·31-s − 0.870·33-s − 0.169·35-s − 0.164·37-s − 0.160·39-s + 1.24·41-s + 0.152·43-s + 0.149·45-s + 0.583·47-s + 1/7·49-s − 0.140·51-s + 0.686·53-s − 0.674·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22848 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22848 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.943719014\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.943719014\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 5 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 9 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 - 15 T + p T^{2} \) |
| 89 | \( 1 - T + p T^{2} \) |
| 97 | \( 1 + 5 T + p T^{2} \) |
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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
−15.55547037768177, −15.02087280430338, −14.30315028193562, −13.71917458838212, −13.47824778793824, −12.94171559859891, −12.24200520846909, −11.85435156935019, −10.97692782998216, −10.31477295533512, −10.06604449084472, −9.447351719443904, −8.876968932362181, −8.148346173388882, −7.634311711678011, −7.297027895776677, −6.187899143894464, −5.943074935194440, −5.127385964312730, −4.465921250704801, −3.761830440026087, −2.869687892990841, −2.467782382515849, −1.739327063987725, −0.5166798537802305,
0.5166798537802305, 1.739327063987725, 2.467782382515849, 2.869687892990841, 3.761830440026087, 4.465921250704801, 5.127385964312730, 5.943074935194440, 6.187899143894464, 7.297027895776677, 7.634311711678011, 8.148346173388882, 8.876968932362181, 9.447351719443904, 10.06604449084472, 10.31477295533512, 10.97692782998216, 11.85435156935019, 12.24200520846909, 12.94171559859891, 13.47824778793824, 13.71917458838212, 14.30315028193562, 15.02087280430338, 15.55547037768177
