| L(s) = 1 | − 2.76·2-s + 5.63·4-s + 1.74·5-s + 4.23·7-s − 10.0·8-s − 4.81·10-s − 11-s + 5.00·13-s − 11.6·14-s + 16.4·16-s − 2.55·17-s − 7.92·19-s + 9.80·20-s + 2.76·22-s + 0.360·23-s − 1.96·25-s − 13.8·26-s + 23.8·28-s − 1.30·29-s + 1.95·31-s − 25.3·32-s + 7.06·34-s + 7.37·35-s − 6.31·37-s + 21.8·38-s − 17.4·40-s − 7.04·41-s + ⋯ |
| L(s) = 1 | − 1.95·2-s + 2.81·4-s + 0.778·5-s + 1.60·7-s − 3.54·8-s − 1.52·10-s − 0.301·11-s + 1.38·13-s − 3.12·14-s + 4.10·16-s − 0.620·17-s − 1.81·19-s + 2.19·20-s + 0.588·22-s + 0.0750·23-s − 0.393·25-s − 2.71·26-s + 4.50·28-s − 0.242·29-s + 0.350·31-s − 4.48·32-s + 1.21·34-s + 1.24·35-s − 1.03·37-s + 3.55·38-s − 2.76·40-s − 1.10·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8019 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8019 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 + 2.76T + 2T^{2} \) |
| 5 | \( 1 - 1.74T + 5T^{2} \) |
| 7 | \( 1 - 4.23T + 7T^{2} \) |
| 13 | \( 1 - 5.00T + 13T^{2} \) |
| 17 | \( 1 + 2.55T + 17T^{2} \) |
| 19 | \( 1 + 7.92T + 19T^{2} \) |
| 23 | \( 1 - 0.360T + 23T^{2} \) |
| 29 | \( 1 + 1.30T + 29T^{2} \) |
| 31 | \( 1 - 1.95T + 31T^{2} \) |
| 37 | \( 1 + 6.31T + 37T^{2} \) |
| 41 | \( 1 + 7.04T + 41T^{2} \) |
| 43 | \( 1 + 5.25T + 43T^{2} \) |
| 47 | \( 1 + 1.52T + 47T^{2} \) |
| 53 | \( 1 + 12.2T + 53T^{2} \) |
| 59 | \( 1 - 3.47T + 59T^{2} \) |
| 61 | \( 1 + 3.92T + 61T^{2} \) |
| 67 | \( 1 + 13.9T + 67T^{2} \) |
| 71 | \( 1 - 15.5T + 71T^{2} \) |
| 73 | \( 1 - 11.2T + 73T^{2} \) |
| 79 | \( 1 + 9.19T + 79T^{2} \) |
| 83 | \( 1 + 1.17T + 83T^{2} \) |
| 89 | \( 1 - 7.31T + 89T^{2} \) |
| 97 | \( 1 + 0.358T + 97T^{2} \) |
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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
−8.022789872853757126367876192633, −6.85941924938974675064139350587, −6.42440055543451589384837258221, −5.73333077189678865377689946952, −4.84315821856373233499197072046, −3.68010403733441065157570765399, −2.47367707669995187691877055067, −1.76275106642943374083842714189, −1.43967645673600346842972472067, 0,
1.43967645673600346842972472067, 1.76275106642943374083842714189, 2.47367707669995187691877055067, 3.68010403733441065157570765399, 4.84315821856373233499197072046, 5.73333077189678865377689946952, 6.42440055543451589384837258221, 6.85941924938974675064139350587, 8.022789872853757126367876192633
