L(s) = 1 | + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s − 1.44·5-s − 0.999·8-s + (−0.724 − 1.25i)10-s − 2·11-s + (2.44 + 4.24i)13-s + (−0.5 − 0.866i)16-s + (−1 − 1.73i)17-s + (1.27 − 2.20i)19-s + (0.724 − 1.25i)20-s + (−1 − 1.73i)22-s + 23-s − 2.89·25-s + (−2.44 + 4.24i)26-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s − 0.648·5-s − 0.353·8-s + (−0.229 − 0.396i)10-s − 0.603·11-s + (0.679 + 1.17i)13-s + (−0.125 − 0.216i)16-s + (−0.242 − 0.420i)17-s + (0.292 − 0.506i)19-s + (0.162 − 0.280i)20-s + (−0.213 − 0.369i)22-s + 0.208·23-s − 0.579·25-s + (−0.480 + 0.832i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.345 + 0.938i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.345 + 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.09570985313\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.09570985313\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + 1.44T + 5T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 + (-2.44 - 4.24i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (1 + 1.73i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.27 + 2.20i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 - T + 23T^{2} \) |
| 29 | \( 1 + (3.44 - 5.97i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-3 + 5.19i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (5.89 - 10.2i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (4.89 + 8.48i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (3.44 - 5.97i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (4.89 + 8.48i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (5.44 + 9.43i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-1 + 1.73i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-3.27 - 5.67i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-6.44 + 11.1i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 0.101T + 71T^{2} \) |
| 73 | \( 1 + (3.44 + 5.97i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.949 - 1.64i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (1 - 1.73i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-8.44 + 14.6i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-1.44 + 2.51i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−8.451538979486441294648626415230, −7.87411036333239082181702843495, −6.90810116133713097036486672946, −6.59113280755628084076120301112, −5.38271771421228505954457631872, −4.79369042744480428501256279366, −3.87134625458873143604900325432, −3.14844539188766627612044599532, −1.80945332153544940004552005197, −0.02795205170989524250949000002,
1.34359875876115205011525392888, 2.60633314597052510229601298366, 3.49274214956498256624178672023, 4.11013448468002165511807373505, 5.20305565190789014759817952038, 5.77764622254140858511673870107, 6.73428768736687625307114219803, 7.900961062962926645008020211931, 8.126765671043835549256837545135, 9.150880919779313672279608531438