| L(s) = 1 | + 2-s + 2·3-s + 4-s − 5-s + 2·6-s + 7-s + 8-s + 9-s − 10-s − 2·11-s + 2·12-s − 13-s + 14-s − 2·15-s + 16-s − 17-s + 18-s − 20-s + 2·21-s − 2·22-s − 6·23-s + 2·24-s + 25-s − 26-s − 4·27-s + 28-s − 4·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.15·3-s + 1/2·4-s − 0.447·5-s + 0.816·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.603·11-s + 0.577·12-s − 0.277·13-s + 0.267·14-s − 0.516·15-s + 1/4·16-s − 0.242·17-s + 0.235·18-s − 0.223·20-s + 0.436·21-s − 0.426·22-s − 1.25·23-s + 0.408·24-s + 1/5·25-s − 0.196·26-s − 0.769·27-s + 0.188·28-s − 0.742·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 88270 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 88270 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.333434780\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.333434780\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 97 | \( 1 - T \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 - 13 T + p T^{2} \) |
| 73 | \( 1 - T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 7 T + p T^{2} \) |
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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
−14.00594397569609, −13.49849365161111, −13.03782214717445, −12.46393150317205, −12.03461839382083, −11.51883181686017, −10.92361049580760, −10.56368125608458, −9.780786370325059, −9.446251213530126, −8.691442762833032, −8.293932661664716, −7.760976855873665, −7.508862374912530, −6.832523751305820, −6.112080290490836, −5.586103440535699, −4.978809833297919, −4.356620199156606, −3.854798370973956, −3.366432907296103, −2.664105856768913, −2.233774180519548, −1.655939711316725, −0.5091029280757753,
0.5091029280757753, 1.655939711316725, 2.233774180519548, 2.664105856768913, 3.366432907296103, 3.854798370973956, 4.356620199156606, 4.978809833297919, 5.586103440535699, 6.112080290490836, 6.832523751305820, 7.508862374912530, 7.760976855873665, 8.293932661664716, 8.691442762833032, 9.446251213530126, 9.780786370325059, 10.56368125608458, 10.92361049580760, 11.51883181686017, 12.03461839382083, 12.46393150317205, 13.03782214717445, 13.49849365161111, 14.00594397569609
