| L(s) = 1 | − 1.89·3-s + 5-s − 7-s + 0.602·9-s − 0.510·11-s + 0.289·13-s − 1.89·15-s − 2.96·17-s + 5.02·19-s + 1.89·21-s + 2.61·23-s + 25-s + 4.55·27-s − 5.87·29-s − 7.31·31-s + 0.969·33-s − 35-s + 3.46·37-s − 0.550·39-s + 3.88·41-s − 43-s + 0.602·45-s − 2.02·47-s + 49-s + 5.62·51-s − 3.69·53-s − 0.510·55-s + ⋯ |
| L(s) = 1 | − 1.09·3-s + 0.447·5-s − 0.377·7-s + 0.200·9-s − 0.153·11-s + 0.0804·13-s − 0.490·15-s − 0.719·17-s + 1.15·19-s + 0.414·21-s + 0.544·23-s + 0.200·25-s + 0.875·27-s − 1.09·29-s − 1.31·31-s + 0.168·33-s − 0.169·35-s + 0.569·37-s − 0.0881·39-s + 0.606·41-s − 0.152·43-s + 0.0897·45-s − 0.295·47-s + 0.142·49-s + 0.787·51-s − 0.507·53-s − 0.0688·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 43 | \( 1 + T \) |
| good | 3 | \( 1 + 1.89T + 3T^{2} \) |
| 11 | \( 1 + 0.510T + 11T^{2} \) |
| 13 | \( 1 - 0.289T + 13T^{2} \) |
| 17 | \( 1 + 2.96T + 17T^{2} \) |
| 19 | \( 1 - 5.02T + 19T^{2} \) |
| 23 | \( 1 - 2.61T + 23T^{2} \) |
| 29 | \( 1 + 5.87T + 29T^{2} \) |
| 31 | \( 1 + 7.31T + 31T^{2} \) |
| 37 | \( 1 - 3.46T + 37T^{2} \) |
| 41 | \( 1 - 3.88T + 41T^{2} \) |
| 47 | \( 1 + 2.02T + 47T^{2} \) |
| 53 | \( 1 + 3.69T + 53T^{2} \) |
| 59 | \( 1 - 8.43T + 59T^{2} \) |
| 61 | \( 1 + 2.16T + 61T^{2} \) |
| 67 | \( 1 + 0.201T + 67T^{2} \) |
| 71 | \( 1 - 11.5T + 71T^{2} \) |
| 73 | \( 1 - 10.1T + 73T^{2} \) |
| 79 | \( 1 - 7.97T + 79T^{2} \) |
| 83 | \( 1 + 15.1T + 83T^{2} \) |
| 89 | \( 1 + 1.93T + 89T^{2} \) |
| 97 | \( 1 + 19.0T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.53345090520155217211168026773, −6.86106627097755949839656654064, −6.26228038889785953855909861222, −5.44331474626372674375141802297, −5.21963532604603234064215751869, −4.14199316809614024839009812826, −3.22974901436227111858215301026, −2.28020098572705414330258704751, −1.12418439297054285149076988193, 0,
1.12418439297054285149076988193, 2.28020098572705414330258704751, 3.22974901436227111858215301026, 4.14199316809614024839009812826, 5.21963532604603234064215751869, 5.44331474626372674375141802297, 6.26228038889785953855909861222, 6.86106627097755949839656654064, 7.53345090520155217211168026773
