| L(s) = 1 | − 2.29·2-s − 0.554·3-s + 3.27·4-s − 0.0860·5-s + 1.27·6-s + 4.43·7-s − 2.92·8-s − 2.69·9-s + 0.197·10-s − 1.81·12-s + 2.58·13-s − 10.1·14-s + 0.0477·15-s + 0.163·16-s + 2.41·17-s + 6.18·18-s + 5.19·19-s − 0.281·20-s − 2.45·21-s + 5.46·23-s + 1.62·24-s − 4.99·25-s − 5.94·26-s + 3.15·27-s + 14.5·28-s − 4.03·29-s − 0.109·30-s + ⋯ |
| L(s) = 1 | − 1.62·2-s − 0.320·3-s + 1.63·4-s − 0.0384·5-s + 0.519·6-s + 1.67·7-s − 1.03·8-s − 0.897·9-s + 0.0624·10-s − 0.523·12-s + 0.717·13-s − 2.72·14-s + 0.0123·15-s + 0.0408·16-s + 0.586·17-s + 1.45·18-s + 1.19·19-s − 0.0629·20-s − 0.536·21-s + 1.14·23-s + 0.330·24-s − 0.998·25-s − 1.16·26-s + 0.607·27-s + 2.74·28-s − 0.749·29-s − 0.0200·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1331 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1331 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8144440390\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8144440390\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 \) |
| good | 2 | \( 1 + 2.29T + 2T^{2} \) |
| 3 | \( 1 + 0.554T + 3T^{2} \) |
| 5 | \( 1 + 0.0860T + 5T^{2} \) |
| 7 | \( 1 - 4.43T + 7T^{2} \) |
| 13 | \( 1 - 2.58T + 13T^{2} \) |
| 17 | \( 1 - 2.41T + 17T^{2} \) |
| 19 | \( 1 - 5.19T + 19T^{2} \) |
| 23 | \( 1 - 5.46T + 23T^{2} \) |
| 29 | \( 1 + 4.03T + 29T^{2} \) |
| 31 | \( 1 - 7.42T + 31T^{2} \) |
| 37 | \( 1 - 0.0124T + 37T^{2} \) |
| 41 | \( 1 + 2.18T + 41T^{2} \) |
| 43 | \( 1 + 1.45T + 43T^{2} \) |
| 47 | \( 1 + 5.67T + 47T^{2} \) |
| 53 | \( 1 + 3.85T + 53T^{2} \) |
| 59 | \( 1 + 11.7T + 59T^{2} \) |
| 61 | \( 1 + 5.99T + 61T^{2} \) |
| 67 | \( 1 - 14.3T + 67T^{2} \) |
| 71 | \( 1 + 5.76T + 71T^{2} \) |
| 73 | \( 1 + 6.81T + 73T^{2} \) |
| 79 | \( 1 - 14.7T + 79T^{2} \) |
| 83 | \( 1 - 15.6T + 83T^{2} \) |
| 89 | \( 1 - 0.837T + 89T^{2} \) |
| 97 | \( 1 + 17.3T + 97T^{2} \) |
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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
−9.461065305550870388881150981246, −8.785729119764906491664581768565, −7.942096258240516067751046921587, −7.76755279800283617087173428813, −6.56490748595927231121410209514, −5.54879826932682080829394287975, −4.74283271687913491052870232762, −3.16576150045199715747392321719, −1.81631582134790401371714770444, −0.909747632554570485407402155901,
0.909747632554570485407402155901, 1.81631582134790401371714770444, 3.16576150045199715747392321719, 4.74283271687913491052870232762, 5.54879826932682080829394287975, 6.56490748595927231121410209514, 7.76755279800283617087173428813, 7.942096258240516067751046921587, 8.785729119764906491664581768565, 9.461065305550870388881150981246
