| L(s) = 1 | + 9.38·2-s + 56.1·4-s + 71.7·5-s + 226.·8-s + 673.·10-s + 560.·11-s + 533.·13-s + 333.·16-s − 1.00e3·17-s + 1.36e3·19-s + 4.02e3·20-s + 5.26e3·22-s − 3.22e3·23-s + 2.02e3·25-s + 5.00e3·26-s + 753.·29-s + 8.20e3·31-s − 4.12e3·32-s − 9.44e3·34-s − 2.80e3·37-s + 1.28e4·38-s + 1.62e4·40-s − 245.·41-s − 1.75e4·43-s + 3.14e4·44-s − 3.03e4·46-s + 1.63e4·47-s + ⋯ |
| L(s) = 1 | + 1.65·2-s + 1.75·4-s + 1.28·5-s + 1.25·8-s + 2.12·10-s + 1.39·11-s + 0.875·13-s + 0.325·16-s − 0.844·17-s + 0.869·19-s + 2.25·20-s + 2.31·22-s − 1.27·23-s + 0.646·25-s + 1.45·26-s + 0.166·29-s + 1.53·31-s − 0.712·32-s − 1.40·34-s − 0.337·37-s + 1.44·38-s + 1.60·40-s − 0.0228·41-s − 1.44·43-s + 2.45·44-s − 2.11·46-s + 1.07·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(8.376419050\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.376419050\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 9.38T + 32T^{2} \) |
| 5 | \( 1 - 71.7T + 3.12e3T^{2} \) |
| 11 | \( 1 - 560.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 533.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.00e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.36e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 3.22e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 753.T + 2.05e7T^{2} \) |
| 31 | \( 1 - 8.20e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 2.80e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 245.T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.75e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.63e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.96e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.03e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 954.T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.98e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 6.21e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.71e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 4.46e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.56e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.36e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.29e4T + 8.58e9T^{2} \) |
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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
−10.48688810960663029427283574015, −9.539026542696432615660624280125, −8.573334065297954853048648904935, −6.89968542849825131146364519967, −6.24497820982057758249024144696, −5.63761477489905406755970060156, −4.48015639875126506976308471848, −3.60120163763069927001350115399, −2.36918199094573995683757906077, −1.36289694726182339144982468708,
1.36289694726182339144982468708, 2.36918199094573995683757906077, 3.60120163763069927001350115399, 4.48015639875126506976308471848, 5.63761477489905406755970060156, 6.24497820982057758249024144696, 6.89968542849825131146364519967, 8.573334065297954853048648904935, 9.539026542696432615660624280125, 10.48688810960663029427283574015
