| L(s) = 1 | + 13.2·5-s + 5.23·7-s + 11-s − 84.9·13-s + 40.9·17-s + 57.5·19-s − 114.·23-s + 50.1·25-s + 202.·29-s − 274.·31-s + 69.2·35-s − 242.·37-s − 328.·41-s + 281.·43-s − 23.8·47-s − 315.·49-s − 300.·53-s + 13.2·55-s + 753.·59-s − 495.·61-s − 1.12e3·65-s − 409.·67-s − 1.11e3·71-s − 287·73-s + 5.23·77-s + 1.23e3·79-s − 942.·83-s + ⋯ |
| L(s) = 1 | + 1.18·5-s + 0.282·7-s + 0.0274·11-s − 1.81·13-s + 0.584·17-s + 0.694·19-s − 1.04·23-s + 0.401·25-s + 1.29·29-s − 1.58·31-s + 0.334·35-s − 1.07·37-s − 1.25·41-s + 0.997·43-s − 0.0740·47-s − 0.920·49-s − 0.777·53-s + 0.0324·55-s + 1.66·59-s − 1.03·61-s − 2.14·65-s − 0.747·67-s − 1.86·71-s − 0.460·73-s + 0.00774·77-s + 1.75·79-s − 1.24·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 13.2T + 125T^{2} \) |
| 7 | \( 1 - 5.23T + 343T^{2} \) |
| 11 | \( 1 - T + 1.33e3T^{2} \) |
| 13 | \( 1 + 84.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 40.9T + 4.91e3T^{2} \) |
| 19 | \( 1 - 57.5T + 6.85e3T^{2} \) |
| 23 | \( 1 + 114.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 202.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 274.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 242.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 328.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 281.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 23.8T + 1.03e5T^{2} \) |
| 53 | \( 1 + 300.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 753.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 495.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 409.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.11e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 287T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.23e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 942.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 190.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 306.T + 9.12e5T^{2} \) |
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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
−8.655245207804427016561027057342, −7.65995581970522236304644594546, −7.04654318111114648694219498127, −6.00919470476714970975290137666, −5.31980992451034609078984818371, −4.64724178061304278728385085557, −3.29989827448718207981552344211, −2.29435112450139826845477479923, −1.52135504784072397550649566365, 0,
1.52135504784072397550649566365, 2.29435112450139826845477479923, 3.29989827448718207981552344211, 4.64724178061304278728385085557, 5.31980992451034609078984818371, 6.00919470476714970975290137666, 7.04654318111114648694219498127, 7.65995581970522236304644594546, 8.655245207804427016561027057342
