L(s) = 1 | + 7.38·3-s + 25·5-s − 49·7-s − 188.·9-s − 226.·11-s − 428.·13-s + 184.·15-s − 315.·17-s + 63.5·19-s − 361.·21-s − 1.12e3·23-s + 625·25-s − 3.18e3·27-s + 634.·29-s − 3.46e3·31-s − 1.67e3·33-s − 1.22e3·35-s − 4.35e3·37-s − 3.16e3·39-s − 9.73e3·41-s + 1.96e3·43-s − 4.71e3·45-s − 1.14e4·47-s + 2.40e3·49-s − 2.33e3·51-s − 7.37e3·53-s − 5.66e3·55-s + ⋯ |
L(s) = 1 | + 0.473·3-s + 0.447·5-s − 0.377·7-s − 0.775·9-s − 0.564·11-s − 0.703·13-s + 0.211·15-s − 0.265·17-s + 0.0404·19-s − 0.178·21-s − 0.445·23-s + 0.200·25-s − 0.840·27-s + 0.139·29-s − 0.646·31-s − 0.267·33-s − 0.169·35-s − 0.522·37-s − 0.333·39-s − 0.904·41-s + 0.161·43-s − 0.346·45-s − 0.754·47-s + 0.142·49-s − 0.125·51-s − 0.360·53-s − 0.252·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - 25T \) |
| 7 | \( 1 + 49T \) |
good | 3 | \( 1 - 7.38T + 243T^{2} \) |
| 11 | \( 1 + 226.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 428.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 315.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 63.5T + 2.47e6T^{2} \) |
| 23 | \( 1 + 1.12e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 634.T + 2.05e7T^{2} \) |
| 31 | \( 1 + 3.46e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 4.35e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 9.73e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.96e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.14e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 7.37e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.47e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.12e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 8.06e3T + 1.35e9T^{2} \) |
| 71 | \( 1 + 2.68e3T + 1.80e9T^{2} \) |
| 73 | \( 1 + 6.33e3T + 2.07e9T^{2} \) |
| 79 | \( 1 - 2.43e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.45e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 9.25e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.04e5T + 8.58e9T^{2} \) |
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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
−11.78757862110369482377627182238, −10.57712219275412756387755138493, −9.593889052339965682158172385552, −8.637776452675241912348686433635, −7.51964385017265384426652435588, −6.18819083616526366692859323519, −5.02038143259769050521257518987, −3.29642065593888974347401724661, −2.14039768334384122017672888240, 0,
2.14039768334384122017672888240, 3.29642065593888974347401724661, 5.02038143259769050521257518987, 6.18819083616526366692859323519, 7.51964385017265384426652435588, 8.637776452675241912348686433635, 9.593889052339965682158172385552, 10.57712219275412756387755138493, 11.78757862110369482377627182238