L(s) = 1 | + (0.5 − 0.866i)2-s + (−0.5 − 0.866i)3-s + (−0.499 − 0.866i)4-s + (0.5 + 0.866i)5-s − 0.999·6-s + (0.5 + 0.866i)7-s − 0.999·8-s + (−0.499 + 0.866i)9-s + 0.999·10-s + 6·11-s + (−0.499 + 0.866i)12-s + (0.5 + 0.866i)13-s + 0.999·14-s + (0.499 − 0.866i)15-s + (−0.5 + 0.866i)16-s + (−3 + 5.19i)17-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (−0.288 − 0.499i)3-s + (−0.249 − 0.433i)4-s + (0.223 + 0.387i)5-s − 0.408·6-s + (0.188 + 0.327i)7-s − 0.353·8-s + (−0.166 + 0.288i)9-s + 0.316·10-s + 1.80·11-s + (−0.144 + 0.249i)12-s + (0.138 + 0.240i)13-s + 0.267·14-s + (0.129 − 0.223i)15-s + (−0.125 + 0.216i)16-s + (−0.727 + 1.26i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.896 + 0.442i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.896 + 0.442i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.943817231\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.943817231\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 + 0.866i)T \) |
| 3 | \( 1 + (0.5 + 0.866i)T \) |
| 5 | \( 1 + (-0.5 - 0.866i)T \) |
| 37 | \( 1 + (-5.5 + 2.59i)T \) |
good | 7 | \( 1 + (-0.5 - 0.866i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 - 6T + 11T^{2} \) |
| 13 | \( 1 + (-0.5 - 0.866i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (3 - 5.19i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.5 - 6.06i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + 6T + 23T^{2} \) |
| 29 | \( 1 - 9T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 41 | \( 1 + (3 + 5.19i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 - 2T + 43T^{2} \) |
| 47 | \( 1 - 12T + 47T^{2} \) |
| 53 | \( 1 + (3 - 5.19i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-3 + 5.19i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (4 + 6.92i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-5 - 8.66i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-1.5 - 2.59i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + 10T + 73T^{2} \) |
| 79 | \( 1 + (7 + 12.1i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-1.5 + 2.59i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 10T + 97T^{2} \) |
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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
−9.953871990082091940938851236081, −9.020732830746475835283969620751, −8.274585891953708611016338931558, −7.09264296644029755282107444231, −6.15886860605771446854277883946, −5.80399760247020061614618320794, −4.28906806490066852181619308797, −3.64903469629494670609858691602, −2.16321645654747343765217437596, −1.35268290978761980872789518326,
0.939947349923226685331139817729, 2.82786302841431761784719495373, 4.15348208551456601175354539965, 4.60502796469245766931162250838, 5.63168780548413473117476710773, 6.54609075013246584393561701979, 7.15395611354107512078693749426, 8.322048092597783534463145228437, 9.228378603034700844243454519974, 9.541927689555402402623443515630