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
−9.987896999323602295944667362561, −8.743826672655527962053513976908, −7.921004076961294836202688532041, −7.33570546745926590939561952219, −6.92017797059270028626300809799, −6.44988161091567980300413380879, −5.35565465703710356263741423772, −4.02617798134232987566453810397, −2.98536851486926970412693212717, −2.20931331483219446306857925897,
0.22898205068513619376748831304, 1.39170354451098520297033575112, 3.02338338272975464266054503340, 3.98841061880335035524970260228, 4.82690640971558012525271893229, 5.44099460122769637995595049255, 6.32161885117300017106737200476, 7.53059225260429499833646654008, 8.623397900773462788185390164045, 8.976433437198697864752956521166