| L(s) = 1 | − 5.40·2-s − 8.99·3-s + 21.2·4-s − 5·5-s + 48.5·6-s + 27.3·7-s − 71.3·8-s + 53.8·9-s + 27.0·10-s − 12.9·11-s − 190.·12-s − 6.73·13-s − 147.·14-s + 44.9·15-s + 215.·16-s + 17·17-s − 291.·18-s − 95.8·19-s − 106.·20-s − 245.·21-s + 69.9·22-s + 41.6·23-s + 641.·24-s + 25·25-s + 36.3·26-s − 241.·27-s + 579.·28-s + ⋯ |
| L(s) = 1 | − 1.91·2-s − 1.73·3-s + 2.65·4-s − 0.447·5-s + 3.30·6-s + 1.47·7-s − 3.15·8-s + 1.99·9-s + 0.854·10-s − 0.354·11-s − 4.58·12-s − 0.143·13-s − 2.82·14-s + 0.773·15-s + 3.37·16-s + 0.242·17-s − 3.81·18-s − 1.15·19-s − 1.18·20-s − 2.55·21-s + 0.678·22-s + 0.377·23-s + 5.45·24-s + 0.200·25-s + 0.274·26-s − 1.72·27-s + 3.91·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 5T \) |
| 17 | \( 1 - 17T \) |
| good | 2 | \( 1 + 5.40T + 8T^{2} \) |
| 3 | \( 1 + 8.99T + 27T^{2} \) |
| 7 | \( 1 - 27.3T + 343T^{2} \) |
| 11 | \( 1 + 12.9T + 1.33e3T^{2} \) |
| 13 | \( 1 + 6.73T + 2.19e3T^{2} \) |
| 19 | \( 1 + 95.8T + 6.85e3T^{2} \) |
| 23 | \( 1 - 41.6T + 1.21e4T^{2} \) |
| 29 | \( 1 - 165.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 197.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 187.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 291.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 139.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 373.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 76.4T + 1.48e5T^{2} \) |
| 59 | \( 1 + 467.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 466.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 206.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 378.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 345.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 194.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 947.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 108.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.69e3T + 9.12e5T^{2} \) |
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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
−12.38975848338446911590952855531, −11.56651722566079551553202921289, −10.90489511485809863749106101463, −10.25493016507439960778135970072, −8.601352264628481923778754897490, −7.58471181274181177873356887188, −6.51009733467806409190166134000, −5.03864484547349625917230657536, −1.54141393018265377577371401791, 0,
1.54141393018265377577371401791, 5.03864484547349625917230657536, 6.51009733467806409190166134000, 7.58471181274181177873356887188, 8.601352264628481923778754897490, 10.25493016507439960778135970072, 10.90489511485809863749106101463, 11.56651722566079551553202921289, 12.38975848338446911590952855531
