| L(s) = 1 | + 14.3·3-s + 25·5-s − 49·7-s − 38.5·9-s − 425.·11-s + 399.·13-s + 357.·15-s + 1.75e3·17-s − 2.87e3·19-s − 700.·21-s + 2.31e3·23-s + 625·25-s − 4.02e3·27-s − 2.12e3·29-s + 1.02e4·31-s − 6.09e3·33-s − 1.22e3·35-s − 7.26e3·37-s + 5.71e3·39-s − 5.89e3·41-s − 2.01e4·43-s − 962.·45-s − 2.00e4·47-s + 2.40e3·49-s + 2.50e4·51-s − 3.39e4·53-s − 1.06e4·55-s + ⋯ |
| L(s) = 1 | + 0.917·3-s + 0.447·5-s − 0.377·7-s − 0.158·9-s − 1.06·11-s + 0.655·13-s + 0.410·15-s + 1.46·17-s − 1.82·19-s − 0.346·21-s + 0.911·23-s + 0.200·25-s − 1.06·27-s − 0.469·29-s + 1.91·31-s − 0.973·33-s − 0.169·35-s − 0.872·37-s + 0.601·39-s − 0.547·41-s − 1.66·43-s − 0.0708·45-s − 1.32·47-s + 0.142·49-s + 1.34·51-s − 1.66·53-s − 0.474·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 14.3T + 243T^{2} \) |
| 11 | \( 1 + 425.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 399.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.75e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.87e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 2.31e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 2.12e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 1.02e4T + 2.86e7T^{2} \) |
| 37 | \( 1 + 7.26e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 5.89e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 2.01e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.00e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.39e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 4.31e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.22e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.75e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 1.65e3T + 1.80e9T^{2} \) |
| 73 | \( 1 + 8.24e3T + 2.07e9T^{2} \) |
| 79 | \( 1 - 9.16e3T + 3.07e9T^{2} \) |
| 83 | \( 1 + 9.52e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.44e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 6.21e4T + 8.58e9T^{2} \) |
| 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
−9.571693969788556555142301050876, −8.370895437128493041019006777810, −8.206356728584180509694023673956, −6.81820668569644275176404756417, −5.89621882924403062431394195558, −4.86359321744990403780691768581, −3.42982930190889232764821188990, −2.77288362971831148770253333890, −1.60199308081565307535669065697, 0,
1.60199308081565307535669065697, 2.77288362971831148770253333890, 3.42982930190889232764821188990, 4.86359321744990403780691768581, 5.89621882924403062431394195558, 6.81820668569644275176404756417, 8.206356728584180509694023673956, 8.370895437128493041019006777810, 9.571693969788556555142301050876
