| L(s) = 1 | − 9.26·2-s + 1.27·3-s + 53.9·4-s + 25·5-s − 11.8·6-s − 203.·8-s − 241.·9-s − 231.·10-s + 478.·11-s + 68.7·12-s − 203.·13-s + 31.8·15-s + 157.·16-s + 854.·17-s + 2.23e3·18-s + 1.38e3·19-s + 1.34e3·20-s − 4.43e3·22-s + 2.41e3·23-s − 258.·24-s + 625·25-s + 1.88e3·26-s − 617.·27-s − 4.20e3·29-s − 295.·30-s − 7.47e3·31-s + 5.03e3·32-s + ⋯ |
| L(s) = 1 | − 1.63·2-s + 0.0817·3-s + 1.68·4-s + 0.447·5-s − 0.133·6-s − 1.12·8-s − 0.993·9-s − 0.732·10-s + 1.19·11-s + 0.137·12-s − 0.333·13-s + 0.0365·15-s + 0.154·16-s + 0.716·17-s + 1.62·18-s + 0.880·19-s + 0.753·20-s − 1.95·22-s + 0.951·23-s − 0.0917·24-s + 0.200·25-s + 0.546·26-s − 0.162·27-s − 0.929·29-s − 0.0599·30-s − 1.39·31-s + 0.869·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.9446242929\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9446242929\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - 25T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 9.26T + 32T^{2} \) |
| 3 | \( 1 - 1.27T + 243T^{2} \) |
| 11 | \( 1 - 478.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 203.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 854.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.38e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 2.41e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 4.20e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 7.47e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 9.59e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 5.50e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 5.02e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.37e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.08e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.04e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 4.41e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 5.31e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 1.58e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 7.76e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 8.61e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 5.21e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 2.32e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.80e5T + 8.58e9T^{2} \) |
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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
−11.19797546322794905858997540895, −9.851109191704134111139141546799, −9.356618744975169476423419585446, −8.531194748543643963601374693242, −7.50872789130056273761278198180, −6.54861384116300954226471973198, −5.33705996232760535499303456249, −3.32097881866655348191528042010, −1.90443712595586434007571851851, −0.74120014221183607482903455362,
0.74120014221183607482903455362, 1.90443712595586434007571851851, 3.32097881866655348191528042010, 5.33705996232760535499303456249, 6.54861384116300954226471973198, 7.50872789130056273761278198180, 8.531194748543643963601374693242, 9.356618744975169476423419585446, 9.851109191704134111139141546799, 11.19797546322794905858997540895
