L(s) = 1 | + (0.0499 + 0.186i)2-s + (1.69 − 0.981i)4-s + (2.35 − 0.632i)7-s + (0.540 + 0.540i)8-s + (2.14 + 1.23i)11-s + (−1.57 − 0.422i)13-s + (0.235 + 0.407i)14-s + (1.88 − 3.27i)16-s + (−0.403 + 0.403i)17-s + 4.28i·19-s + (−0.123 + 0.461i)22-s + (1.82 − 6.82i)23-s − 0.314i·26-s + (3.38 − 3.38i)28-s + (−3.20 + 5.55i)29-s + ⋯ |
L(s) = 1 | + (0.0352 + 0.131i)2-s + (0.849 − 0.490i)4-s + (0.891 − 0.238i)7-s + (0.191 + 0.191i)8-s + (0.646 + 0.373i)11-s + (−0.436 − 0.117i)13-s + (0.0629 + 0.108i)14-s + (0.472 − 0.818i)16-s + (−0.0979 + 0.0979i)17-s + 0.983i·19-s + (−0.0263 + 0.0982i)22-s + (0.381 − 1.42i)23-s − 0.0616i·26-s + (0.640 − 0.640i)28-s + (−0.595 + 1.03i)29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 + 0.241i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.970 + 0.241i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.06864 - 0.253536i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.06864 - 0.253536i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + (-0.0499 - 0.186i)T + (-1.73 + i)T^{2} \) |
| 7 | \( 1 + (-2.35 + 0.632i)T + (6.06 - 3.5i)T^{2} \) |
| 11 | \( 1 + (-2.14 - 1.23i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (1.57 + 0.422i)T + (11.2 + 6.5i)T^{2} \) |
| 17 | \( 1 + (0.403 - 0.403i)T - 17iT^{2} \) |
| 19 | \( 1 - 4.28iT - 19T^{2} \) |
| 23 | \( 1 + (-1.82 + 6.82i)T + (-19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 + (3.20 - 5.55i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (1.97 + 3.41i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.171 - 0.171i)T + 37iT^{2} \) |
| 41 | \( 1 + (-6.52 + 3.76i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (1.32 + 4.95i)T + (-37.2 + 21.5i)T^{2} \) |
| 47 | \( 1 + (-0.780 - 2.91i)T + (-40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (-6.12 - 6.12i)T + 53iT^{2} \) |
| 59 | \( 1 + (2.27 + 3.93i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (0.235 - 0.408i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.443 + 1.65i)T + (-58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 - 3.50iT - 71T^{2} \) |
| 73 | \( 1 + (6.88 - 6.88i)T - 73iT^{2} \) |
| 79 | \( 1 + (-6.50 - 3.75i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (10.6 - 2.85i)T + (71.8 - 41.5i)T^{2} \) |
| 89 | \( 1 - 2.90T + 89T^{2} \) |
| 97 | \( 1 + (1.41 - 0.379i)T + (84.0 - 48.5i)T^{2} \) |
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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
−10.63041909315773869538876711378, −9.740504841550278145399776144126, −8.679387247170815466478592870822, −7.65736934674873341257992780318, −7.00189244305514016761869661355, −6.01325869130161664839978559819, −5.07914970207357331832046604313, −4.01990221643785106728523748110, −2.45061268525804421988124188256, −1.36246924119658506915310848453,
1.55433127574895013178250401122, 2.71708949819116583842294733342, 3.87970262790739554608676960765, 5.04367056680488072678628708756, 6.13359157674987851078957920648, 7.14303868299745092810578264091, 7.79573706175035296135563100741, 8.771626068698913073733774447009, 9.629916960336899558304489337311, 10.83781365465793328927561250112