| L(s) = 1 | + 2.17·2-s + 2.73·4-s + 0.646·5-s − 1.82·7-s + 1.59·8-s + 1.40·10-s − 1.20·11-s − 5.33·13-s − 3.98·14-s − 1.99·16-s + 6.84·17-s + 6.36·19-s + 1.76·20-s − 2.63·22-s − 3.11·23-s − 4.58·25-s − 11.6·26-s − 5.00·28-s − 5.88·29-s − 2.77·31-s − 7.53·32-s + 14.8·34-s − 1.18·35-s − 1.42·37-s + 13.8·38-s + 1.03·40-s + 1.11·41-s + ⋯ |
| L(s) = 1 | + 1.53·2-s + 1.36·4-s + 0.289·5-s − 0.691·7-s + 0.564·8-s + 0.444·10-s − 0.364·11-s − 1.47·13-s − 1.06·14-s − 0.498·16-s + 1.66·17-s + 1.45·19-s + 0.395·20-s − 0.561·22-s − 0.649·23-s − 0.916·25-s − 2.27·26-s − 0.945·28-s − 1.09·29-s − 0.498·31-s − 1.33·32-s + 2.55·34-s − 0.199·35-s − 0.234·37-s + 2.24·38-s + 0.163·40-s + 0.174·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6561 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6561 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 2 | \( 1 - 2.17T + 2T^{2} \) |
| 5 | \( 1 - 0.646T + 5T^{2} \) |
| 7 | \( 1 + 1.82T + 7T^{2} \) |
| 11 | \( 1 + 1.20T + 11T^{2} \) |
| 13 | \( 1 + 5.33T + 13T^{2} \) |
| 17 | \( 1 - 6.84T + 17T^{2} \) |
| 19 | \( 1 - 6.36T + 19T^{2} \) |
| 23 | \( 1 + 3.11T + 23T^{2} \) |
| 29 | \( 1 + 5.88T + 29T^{2} \) |
| 31 | \( 1 + 2.77T + 31T^{2} \) |
| 37 | \( 1 + 1.42T + 37T^{2} \) |
| 41 | \( 1 - 1.11T + 41T^{2} \) |
| 43 | \( 1 + 9.63T + 43T^{2} \) |
| 47 | \( 1 - 6.10T + 47T^{2} \) |
| 53 | \( 1 + 8.52T + 53T^{2} \) |
| 59 | \( 1 - 2.08T + 59T^{2} \) |
| 61 | \( 1 + 3.59T + 61T^{2} \) |
| 67 | \( 1 - 1.21T + 67T^{2} \) |
| 71 | \( 1 - 8.15T + 71T^{2} \) |
| 73 | \( 1 - 6.42T + 73T^{2} \) |
| 79 | \( 1 + 14.2T + 79T^{2} \) |
| 83 | \( 1 - 6.71T + 83T^{2} \) |
| 89 | \( 1 + 15.5T + 89T^{2} \) |
| 97 | \( 1 + 3.72T + 97T^{2} \) |
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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
−7.47122696678769267309603163199, −6.80496486179551685672610946272, −5.89020374573747172215910205136, −5.42692081950948811887300246569, −4.99029310126882587993826902697, −3.91008310216372250119186837548, −3.31164817357293521128530641349, −2.69187803201195236319964362534, −1.71138079494650095685874214551, 0,
1.71138079494650095685874214551, 2.69187803201195236319964362534, 3.31164817357293521128530641349, 3.91008310216372250119186837548, 4.99029310126882587993826902697, 5.42692081950948811887300246569, 5.89020374573747172215910205136, 6.80496486179551685672610946272, 7.47122696678769267309603163199
