| L(s) = 1 | − 2.29·2-s + 1.96·3-s + 3.24·4-s − 1.62·5-s − 4.50·6-s + 3.12·7-s − 2.85·8-s + 0.873·9-s + 3.72·10-s + 4.92·11-s + 6.38·12-s + 1.43·13-s − 7.15·14-s − 3.20·15-s + 0.0397·16-s − 5.08·17-s − 1.99·18-s − 5.38·19-s − 5.28·20-s + 6.14·21-s − 11.2·22-s + 5.97·23-s − 5.61·24-s − 2.34·25-s − 3.29·26-s − 4.18·27-s + 10.1·28-s + ⋯ |
| L(s) = 1 | − 1.61·2-s + 1.13·3-s + 1.62·4-s − 0.728·5-s − 1.84·6-s + 1.18·7-s − 1.00·8-s + 0.291·9-s + 1.17·10-s + 1.48·11-s + 1.84·12-s + 0.399·13-s − 1.91·14-s − 0.827·15-s + 0.00993·16-s − 1.23·17-s − 0.471·18-s − 1.23·19-s − 1.18·20-s + 1.34·21-s − 2.40·22-s + 1.24·23-s − 1.14·24-s − 0.469·25-s − 0.646·26-s − 0.805·27-s + 1.91·28-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
| \(L(1)\) |
\(\approx\) |
\(0.6568991619\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6568991619\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.29T + 2T^{2} \) |
| 3 | \( 1 - 1.96T + 3T^{2} \) |
| 5 | \( 1 + 1.62T + 5T^{2} \) |
| 7 | \( 1 - 3.12T + 7T^{2} \) |
| 11 | \( 1 - 4.92T + 11T^{2} \) |
| 13 | \( 1 - 1.43T + 13T^{2} \) |
| 17 | \( 1 + 5.08T + 17T^{2} \) |
| 19 | \( 1 + 5.38T + 19T^{2} \) |
| 23 | \( 1 - 5.97T + 23T^{2} \) |
| 29 | \( 1 + 7.99T + 29T^{2} \) |
| 31 | \( 1 + 2.39T + 31T^{2} \) |
| 37 | \( 1 - 8.65T + 37T^{2} \) |
| 41 | \( 1 + 4.21T + 41T^{2} \) |
| 43 | \( 1 + 6.23T + 43T^{2} \) |
| 47 | \( 1 + 3.45T + 47T^{2} \) |
| 53 | \( 1 - 12.4T + 53T^{2} \) |
| 59 | \( 1 + 1.60T + 59T^{2} \) |
| 61 | \( 1 + 2.21T + 61T^{2} \) |
| 67 | \( 1 - 10.8T + 67T^{2} \) |
| 71 | \( 1 - 2.67T + 71T^{2} \) |
| 73 | \( 1 - 0.387T + 73T^{2} \) |
| 79 | \( 1 - 0.951T + 79T^{2} \) |
| 89 | \( 1 - 6.39T + 89T^{2} \) |
| 97 | \( 1 + 5.77T + 97T^{2} \) |
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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
−14.82366840336346191034271969360, −13.39088344783023700616725667473, −11.48499520364413823824733216988, −11.07234522203566191378272237575, −9.331332427651280334797921133182, −8.677280350060185718146359472447, −7.999821876836955298078061682949, −6.84179005148419355398924346058, −4.07148209679793956005865039064, −1.90244758485302491746959779534,
1.90244758485302491746959779534, 4.07148209679793956005865039064, 6.84179005148419355398924346058, 7.999821876836955298078061682949, 8.677280350060185718146359472447, 9.331332427651280334797921133182, 11.07234522203566191378272237575, 11.48499520364413823824733216988, 13.39088344783023700616725667473, 14.82366840336346191034271969360
