L(s) = 1 | + (0.5 + 0.866i)2-s + (0.186 − 0.323i)3-s + (−0.499 + 0.866i)4-s + (−1.58 − 2.73i)5-s + 0.373·6-s + (2.36 − 1.18i)7-s − 0.999·8-s + (1.43 + 2.47i)9-s + (1.58 − 2.73i)10-s + (1.05 − 1.83i)11-s + (0.186 + 0.323i)12-s + 2.92·13-s + (2.21 + 1.45i)14-s − 1.18·15-s + (−0.5 − 0.866i)16-s + (3.08 − 5.33i)17-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (0.107 − 0.186i)3-s + (−0.249 + 0.433i)4-s + (−0.707 − 1.22i)5-s + 0.152·6-s + (0.893 − 0.449i)7-s − 0.353·8-s + (0.476 + 0.825i)9-s + (0.500 − 0.866i)10-s + (0.319 − 0.553i)11-s + (0.0539 + 0.0934i)12-s + 0.812·13-s + (0.591 + 0.388i)14-s − 0.305·15-s + (−0.125 − 0.216i)16-s + (0.747 − 1.29i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.120i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 + 0.120i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.60199 - 0.0967384i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.60199 - 0.0967384i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 - 0.866i)T \) |
| 7 | \( 1 + (-2.36 + 1.18i)T \) |
| 23 | \( 1 + (-0.5 - 0.866i)T \) |
good | 3 | \( 1 + (-0.186 + 0.323i)T + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (1.58 + 2.73i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-1.05 + 1.83i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 2.92T + 13T^{2} \) |
| 17 | \( 1 + (-3.08 + 5.33i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.0948 + 0.164i)T + (-9.5 + 16.4i)T^{2} \) |
| 29 | \( 1 + 7.27T + 29T^{2} \) |
| 31 | \( 1 + (2.91 - 5.05i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.69 - 6.40i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 10.6T + 41T^{2} \) |
| 43 | \( 1 + 5.86T + 43T^{2} \) |
| 47 | \( 1 + (-4.45 - 7.70i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (3.56 - 6.16i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-5.61 + 9.72i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (2.45 + 4.25i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (0.573 - 0.993i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 11.8T + 71T^{2} \) |
| 73 | \( 1 + (2.66 - 4.61i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-3.19 - 5.53i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 5.05T + 83T^{2} \) |
| 89 | \( 1 + (-3.90 - 6.76i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 14.7T + 97T^{2} \) |
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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
−11.69746471633842405765561915601, −10.96100424504037229770811094503, −9.456749176340296703232078579751, −8.370100721862759072945431506073, −7.931028453859568697618424822758, −6.96735112761931124652611677192, −5.33889893412971414698061833955, −4.74418173260715118988614443431, −3.61565547089290100557666384613, −1.26669611100894962120646804937,
1.83847671995930325470646366599, 3.49893011166327208093786508804, 4.06397624131547766946806521497, 5.65495167143069387126545729400, 6.76230605127514238164194144041, 7.83804969357247980052614394889, 8.934160031389817274499139030051, 10.07006767020711025663800581219, 10.86600642745871375792894072266, 11.59387181075207028479860191000