L(s) = 1 | + (−0.382 − 0.662i)2-s + (−0.5 + 0.866i)3-s + (0.207 − 0.358i)4-s + (−0.923 − 1.60i)5-s + 0.765·6-s − 1.08·8-s + (−0.499 − 0.866i)9-s + (−0.707 + 1.22i)10-s + (0.382 − 0.662i)11-s + (0.207 + 0.358i)12-s − 13-s + 1.84·15-s + (0.207 + 0.358i)16-s + (−0.382 + 0.662i)18-s − 0.765·20-s + ⋯ |
L(s) = 1 | + (−0.382 − 0.662i)2-s + (−0.5 + 0.866i)3-s + (0.207 − 0.358i)4-s + (−0.923 − 1.60i)5-s + 0.765·6-s − 1.08·8-s + (−0.499 − 0.866i)9-s + (−0.707 + 1.22i)10-s + (0.382 − 0.662i)11-s + (0.207 + 0.358i)12-s − 13-s + 1.84·15-s + (0.207 + 0.358i)16-s + (−0.382 + 0.662i)18-s − 0.765·20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.749 - 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.749 - 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2575176272\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2575176272\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.5 - 0.866i)T \) |
| 7 | \( 1 \) |
| 13 | \( 1 + T \) |
good | 2 | \( 1 + (0.382 + 0.662i)T + (-0.5 + 0.866i)T^{2} \) |
| 5 | \( 1 + (0.923 + 1.60i)T + (-0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.382 + 0.662i)T + (-0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 + 1.84T + T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + (-0.923 - 1.60i)T + (-0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + (-0.382 + 0.662i)T + (-0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + 1.84T + T^{2} \) |
| 73 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 - 0.765T + T^{2} \) |
| 89 | \( 1 + (0.382 + 0.662i)T + (-0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−8.976433437198697864752956521166, −8.623397900773462788185390164045, −7.53059225260429499833646654008, −6.32161885117300017106737200476, −5.44099460122769637995595049255, −4.82690640971558012525271893229, −3.98841061880335035524970260228, −3.02338338272975464266054503340, −1.39170354451098520297033575112, −0.22898205068513619376748831304,
2.20931331483219446306857925897, 2.98536851486926970412693212717, 4.02617798134232987566453810397, 5.35565465703710356263741423772, 6.44988161091567980300413380879, 6.92017797059270028626300809799, 7.33570546745926590939561952219, 7.921004076961294836202688532041, 8.743826672655527962053513976908, 9.987896999323602295944667362561