| L(s) = 1 | − 8.02e12·2-s + 1.02e20·3-s + 2.56e25·4-s + 8.73e29·5-s − 8.25e32·6-s + 1.26e35·7-s + 1.04e38·8-s − 2.53e40·9-s − 7.00e42·10-s + 5.36e42·11-s + 2.63e45·12-s − 2.69e47·13-s − 1.01e48·14-s + 8.98e49·15-s − 1.83e51·16-s − 1.84e52·17-s + 2.03e53·18-s − 1.42e54·19-s + 2.23e55·20-s + 1.29e55·21-s − 4.30e55·22-s + 1.16e58·23-s + 1.07e58·24-s + 5.04e59·25-s + 2.16e60·26-s − 6.30e60·27-s + 3.23e60·28-s + ⋯ |
| L(s) = 1 | − 1.28·2-s + 0.542·3-s + 0.662·4-s + 1.71·5-s − 0.700·6-s + 0.152·7-s + 0.434·8-s − 0.705·9-s − 2.21·10-s + 0.0295·11-s + 0.359·12-s − 1.22·13-s − 0.197·14-s + 0.932·15-s − 1.22·16-s − 0.939·17-s + 0.909·18-s − 0.642·19-s + 1.13·20-s + 0.0829·21-s − 0.0380·22-s + 1.55·23-s + 0.236·24-s + 1.95·25-s + 1.58·26-s − 0.925·27-s + 0.101·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & -\,\Lambda(86-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+85/2) \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(43)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{87}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| good | 2 | \( 1 + 8.02e12T + 3.86e25T^{2} \) |
| 3 | \( 1 - 1.02e20T + 3.59e40T^{2} \) |
| 5 | \( 1 - 8.73e29T + 2.58e59T^{2} \) |
| 7 | \( 1 - 1.26e35T + 6.81e71T^{2} \) |
| 11 | \( 1 - 5.36e42T + 3.29e88T^{2} \) |
| 13 | \( 1 + 2.69e47T + 4.84e94T^{2} \) |
| 17 | \( 1 + 1.84e52T + 3.87e104T^{2} \) |
| 19 | \( 1 + 1.42e54T + 4.94e108T^{2} \) |
| 23 | \( 1 - 1.16e58T + 5.58e115T^{2} \) |
| 29 | \( 1 + 1.62e61T + 2.01e124T^{2} \) |
| 31 | \( 1 + 2.67e63T + 5.83e126T^{2} \) |
| 37 | \( 1 + 7.87e66T + 1.98e133T^{2} \) |
| 41 | \( 1 - 1.54e68T + 1.22e137T^{2} \) |
| 43 | \( 1 + 3.44e69T + 6.99e138T^{2} \) |
| 47 | \( 1 - 1.94e71T + 1.34e142T^{2} \) |
| 53 | \( 1 + 2.32e73T + 3.65e146T^{2} \) |
| 59 | \( 1 - 2.41e75T + 3.32e150T^{2} \) |
| 61 | \( 1 - 3.93e75T + 5.66e151T^{2} \) |
| 67 | \( 1 + 1.23e77T + 1.64e155T^{2} \) |
| 71 | \( 1 + 3.56e78T + 2.27e157T^{2} \) |
| 73 | \( 1 + 2.16e78T + 2.41e158T^{2} \) |
| 79 | \( 1 + 3.82e80T + 1.98e161T^{2} \) |
| 83 | \( 1 - 3.91e81T + 1.32e163T^{2} \) |
| 89 | \( 1 + 3.41e81T + 4.99e165T^{2} \) |
| 97 | \( 1 + 2.02e84T + 7.50e168T^{2} \) |
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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
−14.56148109612782450919178556915, −13.27631754686104045127952239038, −10.73399067079389547215894963502, −9.463933219523116695451670193963, −8.713926760865402391432012203976, −6.93233153668210653722590023123, −5.16152845667467136105999056368, −2.53000443542404226135630695713, −1.67820578476657413249869961083, 0,
1.67820578476657413249869961083, 2.53000443542404226135630695713, 5.16152845667467136105999056368, 6.93233153668210653722590023123, 8.713926760865402391432012203976, 9.463933219523116695451670193963, 10.73399067079389547215894963502, 13.27631754686104045127952239038, 14.56148109612782450919178556915
