| L(s) = 1 | − 10.2·2-s − 61.4·3-s − 407.·4-s − 1.44e3·5-s + 629.·6-s − 2.40e3·7-s + 9.41e3·8-s − 1.59e4·9-s + 1.48e4·10-s + 8.17e4·11-s + 2.49e4·12-s − 2.85e4·13-s + 2.46e4·14-s + 8.90e4·15-s + 1.11e5·16-s + 4.20e5·17-s + 1.63e5·18-s − 2.47e5·19-s + 5.89e5·20-s + 1.47e5·21-s − 8.38e5·22-s + 1.73e5·23-s − 5.78e5·24-s + 1.48e5·25-s + 2.92e5·26-s + 2.18e6·27-s + 9.77e5·28-s + ⋯ |
| L(s) = 1 | − 0.452·2-s − 0.437·3-s − 0.794·4-s − 1.03·5-s + 0.198·6-s − 0.377·7-s + 0.812·8-s − 0.808·9-s + 0.469·10-s + 1.68·11-s + 0.347·12-s − 0.277·13-s + 0.171·14-s + 0.453·15-s + 0.426·16-s + 1.22·17-s + 0.366·18-s − 0.436·19-s + 0.824·20-s + 0.165·21-s − 0.762·22-s + 0.129·23-s − 0.355·24-s + 0.0758·25-s + 0.125·26-s + 0.791·27-s + 0.300·28-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 | \( 1 + 10.2T + 512T^{2} \) |
| 3 | \( 1 + 61.4T + 1.96e4T^{2} \) |
| 5 | \( 1 + 1.44e3T + 1.95e6T^{2} \) |
| 11 | \( 1 - 8.17e4T + 2.35e9T^{2} \) |
| 17 | \( 1 - 4.20e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 2.47e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 1.73e5T + 1.80e12T^{2} \) |
| 29 | \( 1 - 4.14e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 3.12e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.06e7T + 1.29e14T^{2} \) |
| 41 | \( 1 - 1.89e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 4.40e6T + 5.02e14T^{2} \) |
| 47 | \( 1 - 4.89e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 4.97e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 6.54e6T + 8.66e15T^{2} \) |
| 61 | \( 1 + 1.65e8T + 1.16e16T^{2} \) |
| 67 | \( 1 - 8.11e7T + 2.72e16T^{2} \) |
| 71 | \( 1 + 2.87e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 6.04e7T + 5.88e16T^{2} \) |
| 79 | \( 1 + 4.42e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 1.80e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 6.84e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 3.24e8T + 7.60e17T^{2} \) |
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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
−11.82842298733643905784209909481, −10.62216097604319536635456558663, −9.356833841116361751817033530707, −8.509764052139860568453468040856, −7.34902262499201032335224529095, −5.92286035347018607460942003711, −4.44955557808259880414247822613, −3.42501196761626769003631379914, −1.05147745545596536232418253791, 0,
1.05147745545596536232418253791, 3.42501196761626769003631379914, 4.44955557808259880414247822613, 5.92286035347018607460942003711, 7.34902262499201032335224529095, 8.509764052139860568453468040856, 9.356833841116361751817033530707, 10.62216097604319536635456558663, 11.82842298733643905784209909481
