| L(s) = 1 | + 3.55e11·2-s + 1.22e18·3-s − 2.50e22·4-s + 1.38e27·5-s + 4.36e29·6-s − 5.00e32·7-s − 6.25e34·8-s − 3.96e36·9-s + 4.92e38·10-s − 1.21e40·11-s − 3.07e40·12-s − 3.54e41·13-s − 1.77e44·14-s + 1.70e45·15-s − 1.84e46·16-s + 1.84e47·17-s − 1.40e48·18-s − 2.35e49·19-s − 3.46e49·20-s − 6.14e50·21-s − 4.31e51·22-s + 1.55e52·23-s − 7.69e52·24-s + 1.25e54·25-s − 1.26e53·26-s − 1.16e55·27-s + 1.25e55·28-s + ⋯ |
| L(s) = 1 | + 0.913·2-s + 0.525·3-s − 0.165·4-s + 1.70·5-s + 0.480·6-s − 1.45·7-s − 1.06·8-s − 0.723·9-s + 1.55·10-s − 0.979·11-s − 0.0869·12-s − 0.0460·13-s − 1.32·14-s + 0.895·15-s − 0.807·16-s + 0.781·17-s − 0.661·18-s − 1.37·19-s − 0.281·20-s − 0.764·21-s − 0.894·22-s + 0.582·23-s − 0.559·24-s + 1.90·25-s − 0.0420·26-s − 0.905·27-s + 0.240·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & -\,\Lambda(78-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+77/2) \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(39)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{79}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| good | 2 | \( 1 - 3.55e11T + 1.51e23T^{2} \) |
| 3 | \( 1 - 1.22e18T + 5.47e36T^{2} \) |
| 5 | \( 1 - 1.38e27T + 6.61e53T^{2} \) |
| 7 | \( 1 + 5.00e32T + 1.18e65T^{2} \) |
| 11 | \( 1 + 1.21e40T + 1.53e80T^{2} \) |
| 13 | \( 1 + 3.54e41T + 5.93e85T^{2} \) |
| 17 | \( 1 - 1.84e47T + 5.55e94T^{2} \) |
| 19 | \( 1 + 2.35e49T + 2.91e98T^{2} \) |
| 23 | \( 1 - 1.55e52T + 7.12e104T^{2} \) |
| 29 | \( 1 + 2.36e56T + 4.02e112T^{2} \) |
| 31 | \( 1 - 1.62e56T + 6.83e114T^{2} \) |
| 37 | \( 1 - 1.28e60T + 5.64e120T^{2} \) |
| 41 | \( 1 + 1.91e62T + 1.52e124T^{2} \) |
| 43 | \( 1 + 3.74e62T + 5.98e125T^{2} \) |
| 47 | \( 1 - 6.85e63T + 5.64e128T^{2} \) |
| 53 | \( 1 - 5.42e65T + 5.87e132T^{2} \) |
| 59 | \( 1 + 7.28e67T + 2.26e136T^{2} \) |
| 61 | \( 1 - 5.56e68T + 2.95e137T^{2} \) |
| 67 | \( 1 + 1.72e70T + 4.05e140T^{2} \) |
| 71 | \( 1 - 2.59e71T + 3.52e142T^{2} \) |
| 73 | \( 1 + 9.27e71T + 2.99e143T^{2} \) |
| 79 | \( 1 + 6.11e72T + 1.31e146T^{2} \) |
| 83 | \( 1 - 1.75e73T + 5.87e147T^{2} \) |
| 89 | \( 1 - 1.53e75T + 1.26e150T^{2} \) |
| 97 | \( 1 + 4.42e75T + 9.58e152T^{2} \) |
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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
−14.81684674503952522943008750881, −13.48673968521877326341246397424, −12.89741676239236568232492801686, −10.04657034162099175986649163573, −8.942047150104341786171008785629, −6.26793333844939544423092155334, −5.37208176044697332874969272562, −3.28996470810543097775428955450, −2.36854966842760420614828607236, 0,
2.36854966842760420614828607236, 3.28996470810543097775428955450, 5.37208176044697332874969272562, 6.26793333844939544423092155334, 8.942047150104341786171008785629, 10.04657034162099175986649163573, 12.89741676239236568232492801686, 13.48673968521877326341246397424, 14.81684674503952522943008750881
