L(s) = 1 | + 3-s + 5-s + 9-s + 4·11-s + 15-s − 6·17-s + 4·19-s − 23-s + 25-s + 27-s − 6·29-s − 4·31-s + 4·33-s + 2·37-s + 4·41-s + 10·43-s + 45-s + 6·47-s − 6·51-s − 2·53-s + 4·55-s + 4·57-s − 2·59-s − 2·61-s + 10·67-s − 69-s + 6·73-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.447·5-s + 1/3·9-s + 1.20·11-s + 0.258·15-s − 1.45·17-s + 0.917·19-s − 0.208·23-s + 1/5·25-s + 0.192·27-s − 1.11·29-s − 0.718·31-s + 0.696·33-s + 0.328·37-s + 0.624·41-s + 1.52·43-s + 0.149·45-s + 0.875·47-s − 0.840·51-s − 0.274·53-s + 0.539·55-s + 0.529·57-s − 0.260·59-s − 0.256·61-s + 1.22·67-s − 0.120·69-s + 0.702·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 135240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.062919055\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.062919055\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| 23 | \( 1 + T \) |
good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−13.60005564009160, −13.04848550937214, −12.49280658131235, −12.14127278019446, −11.41279183692250, −11.02609027890249, −10.70352748153084, −9.811220046603601, −9.475130565216434, −9.110592713609777, −8.794249029976375, −8.085161011561097, −7.437218675001547, −7.161863118820577, −6.476312687663319, −6.069501260600512, −5.482760633990166, −4.834120403365566, −4.138684669065429, −3.861018748793424, −3.181468494635750, −2.399603733234427, −2.027110856440600, −1.311677813416558, −0.5920785091469214,
0.5920785091469214, 1.311677813416558, 2.027110856440600, 2.399603733234427, 3.181468494635750, 3.861018748793424, 4.138684669065429, 4.834120403365566, 5.482760633990166, 6.069501260600512, 6.476312687663319, 7.161863118820577, 7.437218675001547, 8.085161011561097, 8.794249029976375, 9.110592713609777, 9.475130565216434, 9.811220046603601, 10.70352748153084, 11.02609027890249, 11.41279183692250, 12.14127278019446, 12.49280658131235, 13.04848550937214, 13.60005564009160