| L(s) = 1 | − 139.·2-s − 6.06e3·3-s − 1.33e4·4-s + 2.45e5·5-s + 8.44e5·6-s + 3.71e6·7-s + 6.42e6·8-s + 2.24e7·9-s − 3.41e7·10-s + 5.32e7·11-s + 8.10e7·12-s − 1.96e8·13-s − 5.17e8·14-s − 1.48e9·15-s − 4.57e8·16-s + 3.11e9·17-s − 3.12e9·18-s − 6.76e9·19-s − 3.27e9·20-s − 2.25e10·21-s − 7.41e9·22-s − 2.91e9·23-s − 3.89e10·24-s + 2.95e10·25-s + 2.74e10·26-s − 4.90e10·27-s − 4.96e10·28-s + ⋯ |
| L(s) = 1 | − 0.769·2-s − 1.60·3-s − 0.407·4-s + 1.40·5-s + 1.23·6-s + 1.70·7-s + 1.08·8-s + 1.56·9-s − 1.07·10-s + 0.823·11-s + 0.652·12-s − 0.869·13-s − 1.31·14-s − 2.24·15-s − 0.425·16-s + 1.84·17-s − 1.20·18-s − 1.73·19-s − 0.572·20-s − 2.72·21-s − 0.633·22-s − 0.178·23-s − 1.73·24-s + 0.967·25-s + 0.669·26-s − 0.902·27-s − 0.694·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(8)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{17}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 83 | \( 1 - 2.71e13T \) |
| good | 2 | \( 1 + 139.T + 3.27e4T^{2} \) |
| 3 | \( 1 + 6.06e3T + 1.43e7T^{2} \) |
| 5 | \( 1 - 2.45e5T + 3.05e10T^{2} \) |
| 7 | \( 1 - 3.71e6T + 4.74e12T^{2} \) |
| 11 | \( 1 - 5.32e7T + 4.17e15T^{2} \) |
| 13 | \( 1 + 1.96e8T + 5.11e16T^{2} \) |
| 17 | \( 1 - 3.11e9T + 2.86e18T^{2} \) |
| 19 | \( 1 + 6.76e9T + 1.51e19T^{2} \) |
| 23 | \( 1 + 2.91e9T + 2.66e20T^{2} \) |
| 29 | \( 1 + 1.51e10T + 8.62e21T^{2} \) |
| 31 | \( 1 - 2.65e10T + 2.34e22T^{2} \) |
| 37 | \( 1 + 5.31e11T + 3.33e23T^{2} \) |
| 41 | \( 1 + 2.17e12T + 1.55e24T^{2} \) |
| 43 | \( 1 - 2.88e12T + 3.17e24T^{2} \) |
| 47 | \( 1 + 4.96e12T + 1.20e25T^{2} \) |
| 53 | \( 1 + 3.92e12T + 7.31e25T^{2} \) |
| 59 | \( 1 + 5.53e12T + 3.65e26T^{2} \) |
| 61 | \( 1 + 3.76e13T + 6.02e26T^{2} \) |
| 67 | \( 1 + 4.01e13T + 2.46e27T^{2} \) |
| 71 | \( 1 + 1.10e14T + 5.87e27T^{2} \) |
| 73 | \( 1 + 4.14e13T + 8.90e27T^{2} \) |
| 79 | \( 1 + 2.38e14T + 2.91e28T^{2} \) |
| 89 | \( 1 + 2.24e14T + 1.74e29T^{2} \) |
| 97 | \( 1 + 9.43e12T + 6.33e29T^{2} \) |
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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
−10.51990646909412004055368335578, −10.03549069609760184989091666097, −8.762793439927409093143798172089, −7.49129727757127415087304628048, −6.11358564108397483556936314270, −5.19503562314832863212247673946, −4.51204619105910316353629406827, −1.70934353338162940342060769308, −1.32031399293420341954670211554, 0,
1.32031399293420341954670211554, 1.70934353338162940342060769308, 4.51204619105910316353629406827, 5.19503562314832863212247673946, 6.11358564108397483556936314270, 7.49129727757127415087304628048, 8.762793439927409093143798172089, 10.03549069609760184989091666097, 10.51990646909412004055368335578
