| L(s) = 1 | − 39.6·2-s − 201.·3-s + 1.06e3·4-s + 764.·5-s + 8.00e3·6-s + 2.40e3·7-s − 2.18e4·8-s + 2.10e4·9-s − 3.03e4·10-s − 8.70e4·11-s − 2.14e5·12-s + 2.85e4·13-s − 9.52e4·14-s − 1.54e5·15-s + 3.22e5·16-s − 4.61e4·17-s − 8.35e5·18-s + 5.04e5·19-s + 8.12e5·20-s − 4.84e5·21-s + 3.45e6·22-s − 5.99e5·23-s + 4.40e6·24-s − 1.36e6·25-s − 1.13e6·26-s − 2.76e5·27-s + 2.55e6·28-s + ⋯ |
| L(s) = 1 | − 1.75·2-s − 1.43·3-s + 2.07·4-s + 0.546·5-s + 2.52·6-s + 0.377·7-s − 1.88·8-s + 1.06·9-s − 0.959·10-s − 1.79·11-s − 2.98·12-s + 0.277·13-s − 0.662·14-s − 0.786·15-s + 1.23·16-s − 0.134·17-s − 1.87·18-s + 0.888·19-s + 1.13·20-s − 0.543·21-s + 3.14·22-s − 0.446·23-s + 2.71·24-s − 0.700·25-s − 0.486·26-s − 0.100·27-s + 0.784·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - 2.40e3T \) |
| 13 | \( 1 - 2.85e4T \) |
| good | 2 | \( 1 + 39.6T + 512T^{2} \) |
| 3 | \( 1 + 201.T + 1.96e4T^{2} \) |
| 5 | \( 1 - 764.T + 1.95e6T^{2} \) |
| 11 | \( 1 + 8.70e4T + 2.35e9T^{2} \) |
| 17 | \( 1 + 4.61e4T + 1.18e11T^{2} \) |
| 19 | \( 1 - 5.04e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 5.99e5T + 1.80e12T^{2} \) |
| 29 | \( 1 - 2.58e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 3.96e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 6.19e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + 1.66e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 3.31e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 2.64e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 3.90e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 4.81e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 1.43e8T + 1.16e16T^{2} \) |
| 67 | \( 1 + 1.77e7T + 2.72e16T^{2} \) |
| 71 | \( 1 + 8.31e7T + 4.58e16T^{2} \) |
| 73 | \( 1 - 2.29e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 2.99e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 8.34e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 2.13e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 1.36e9T + 7.60e17T^{2} \) |
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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
−11.25802365071237522677960872965, −10.48288403767470548980079761797, −9.852023700021116824801156792254, −8.367446488467166015682593224959, −7.37061794508092070714796371796, −6.11604079431850129466228299444, −5.14335896887931214894494143410, −2.38465508933771413308684296504, −1.01677652399347528943642336579, 0,
1.01677652399347528943642336579, 2.38465508933771413308684296504, 5.14335896887931214894494143410, 6.11604079431850129466228299444, 7.37061794508092070714796371796, 8.367446488467166015682593224959, 9.852023700021116824801156792254, 10.48288403767470548980079761797, 11.25802365071237522677960872965
