| L(s) = 1 | − 2.45·3-s + 1.53·5-s + 1.05·7-s + 3.00·9-s + 2.99·11-s − 0.170·13-s − 3.75·15-s + 3.06·17-s − 19-s − 2.58·21-s − 8.77·23-s − 2.65·25-s − 0.0239·27-s − 5.85·29-s + 9.59·31-s − 7.33·33-s + 1.61·35-s + 2.36·37-s + 0.418·39-s − 6.12·41-s + 4.52·43-s + 4.60·45-s − 9.15·47-s − 5.89·49-s − 7.51·51-s − 11.9·53-s + 4.57·55-s + ⋯ |
| L(s) = 1 | − 1.41·3-s + 0.684·5-s + 0.397·7-s + 1.00·9-s + 0.902·11-s − 0.0473·13-s − 0.968·15-s + 0.743·17-s − 0.229·19-s − 0.563·21-s − 1.82·23-s − 0.531·25-s − 0.00461·27-s − 1.08·29-s + 1.72·31-s − 1.27·33-s + 0.272·35-s + 0.388·37-s + 0.0669·39-s − 0.955·41-s + 0.690·43-s + 0.686·45-s − 1.33·47-s − 0.841·49-s − 1.05·51-s − 1.64·53-s + 0.617·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 19 | \( 1 + T \) |
| 79 | \( 1 - T \) |
| good | 3 | \( 1 + 2.45T + 3T^{2} \) |
| 5 | \( 1 - 1.53T + 5T^{2} \) |
| 7 | \( 1 - 1.05T + 7T^{2} \) |
| 11 | \( 1 - 2.99T + 11T^{2} \) |
| 13 | \( 1 + 0.170T + 13T^{2} \) |
| 17 | \( 1 - 3.06T + 17T^{2} \) |
| 23 | \( 1 + 8.77T + 23T^{2} \) |
| 29 | \( 1 + 5.85T + 29T^{2} \) |
| 31 | \( 1 - 9.59T + 31T^{2} \) |
| 37 | \( 1 - 2.36T + 37T^{2} \) |
| 41 | \( 1 + 6.12T + 41T^{2} \) |
| 43 | \( 1 - 4.52T + 43T^{2} \) |
| 47 | \( 1 + 9.15T + 47T^{2} \) |
| 53 | \( 1 + 11.9T + 53T^{2} \) |
| 59 | \( 1 + 4.41T + 59T^{2} \) |
| 61 | \( 1 + 9.85T + 61T^{2} \) |
| 67 | \( 1 - 5.05T + 67T^{2} \) |
| 71 | \( 1 - 2.90T + 71T^{2} \) |
| 73 | \( 1 - 2.32T + 73T^{2} \) |
| 83 | \( 1 + 7.65T + 83T^{2} \) |
| 89 | \( 1 + 2.18T + 89T^{2} \) |
| 97 | \( 1 - 10.0T + 97T^{2} \) |
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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
−7.74486843583058439580137350824, −6.66953661736813300036942170477, −6.15139619875180352124780520617, −5.80232163796313636836451122498, −4.91214739795042436350219487504, −4.32710691930775214370283293600, −3.33561913923750715640553908950, −1.98306693119650155479326381980, −1.29139741741965716941342862506, 0,
1.29139741741965716941342862506, 1.98306693119650155479326381980, 3.33561913923750715640553908950, 4.32710691930775214370283293600, 4.91214739795042436350219487504, 5.80232163796313636836451122498, 6.15139619875180352124780520617, 6.66953661736813300036942170477, 7.74486843583058439580137350824
