| L(s) = 1 | + 2.62·2-s − 1.49·3-s + 4.88·4-s − 3.47·5-s − 3.92·6-s − 1.39·7-s + 7.56·8-s − 0.762·9-s − 9.11·10-s + 1.06·11-s − 7.30·12-s + 6.94·13-s − 3.66·14-s + 5.19·15-s + 10.0·16-s − 1.62·17-s − 2.00·18-s − 2.37·19-s − 16.9·20-s + 2.09·21-s + 2.78·22-s − 5.23·23-s − 11.3·24-s + 7.06·25-s + 18.2·26-s + 5.62·27-s − 6.82·28-s + ⋯ |
| L(s) = 1 | + 1.85·2-s − 0.863·3-s + 2.44·4-s − 1.55·5-s − 1.60·6-s − 0.528·7-s + 2.67·8-s − 0.254·9-s − 2.88·10-s + 0.320·11-s − 2.10·12-s + 1.92·13-s − 0.980·14-s + 1.34·15-s + 2.51·16-s − 0.394·17-s − 0.471·18-s − 0.544·19-s − 3.79·20-s + 0.456·21-s + 0.594·22-s − 1.09·23-s − 2.30·24-s + 1.41·25-s + 3.57·26-s + 1.08·27-s − 1.29·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.629820907\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.629820907\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 83 | \( 1 - T \) |
| good | 2 | \( 1 - 2.62T + 2T^{2} \) |
| 3 | \( 1 + 1.49T + 3T^{2} \) |
| 5 | \( 1 + 3.47T + 5T^{2} \) |
| 7 | \( 1 + 1.39T + 7T^{2} \) |
| 11 | \( 1 - 1.06T + 11T^{2} \) |
| 13 | \( 1 - 6.94T + 13T^{2} \) |
| 17 | \( 1 + 1.62T + 17T^{2} \) |
| 19 | \( 1 + 2.37T + 19T^{2} \) |
| 23 | \( 1 + 5.23T + 23T^{2} \) |
| 29 | \( 1 - 1.37T + 29T^{2} \) |
| 31 | \( 1 + 2.98T + 31T^{2} \) |
| 37 | \( 1 - 5.84T + 37T^{2} \) |
| 41 | \( 1 - 2.13T + 41T^{2} \) |
| 43 | \( 1 + 6.24T + 43T^{2} \) |
| 47 | \( 1 - 1.67T + 47T^{2} \) |
| 53 | \( 1 - 3.44T + 53T^{2} \) |
| 59 | \( 1 + 14.2T + 59T^{2} \) |
| 61 | \( 1 - 6.40T + 61T^{2} \) |
| 67 | \( 1 - 6.41T + 67T^{2} \) |
| 71 | \( 1 - 2.29T + 71T^{2} \) |
| 73 | \( 1 - 11.6T + 73T^{2} \) |
| 79 | \( 1 + 10.7T + 79T^{2} \) |
| 89 | \( 1 + 13.6T + 89T^{2} \) |
| 97 | \( 1 + 3.25T + 97T^{2} \) |
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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
−14.22934602991949551648998634063, −13.08816053194587198599749411384, −12.19164652852055333265765103099, −11.38186492502161595168038735362, −10.89769103117052955770726745309, −8.258453357251033047121093677687, −6.68980341009942874659135231730, −5.91044245171285488749258939503, −4.34642816296284733164774881667, −3.46210416166494792970647632312,
3.46210416166494792970647632312, 4.34642816296284733164774881667, 5.91044245171285488749258939503, 6.68980341009942874659135231730, 8.258453357251033047121093677687, 10.89769103117052955770726745309, 11.38186492502161595168038735362, 12.19164652852055333265765103099, 13.08816053194587198599749411384, 14.22934602991949551648998634063
