L(s) = 1 | + (0.5 − 0.866i)3-s + 5-s + (−0.5 − 0.866i)7-s + (−0.499 − 0.866i)9-s + (2.13 − 3.70i)11-s + (2.5 − 2.59i)13-s + (0.5 − 0.866i)15-s + (1.13 + 1.97i)19-s − 0.999·21-s + (−3.27 + 5.67i)23-s + 25-s − 0.999·27-s + (5.27 − 9.13i)29-s − 7.27·31-s + (−2.13 − 3.70i)33-s + ⋯ |
L(s) = 1 | + (0.288 − 0.499i)3-s + 0.447·5-s + (−0.188 − 0.327i)7-s + (−0.166 − 0.288i)9-s + (0.644 − 1.11i)11-s + (0.693 − 0.720i)13-s + (0.129 − 0.223i)15-s + (0.260 + 0.451i)19-s − 0.218·21-s + (−0.682 + 1.18i)23-s + 0.200·25-s − 0.192·27-s + (0.979 − 1.69i)29-s − 1.30·31-s + (−0.372 − 0.644i)33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0128 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0128 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.976270264\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.976270264\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.5 + 0.866i)T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 + (-2.5 + 2.59i)T \) |
good | 7 | \( 1 + (0.5 + 0.866i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-2.13 + 3.70i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.13 - 1.97i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (3.27 - 5.67i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-5.27 + 9.13i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 7.27T + 31T^{2} \) |
| 37 | \( 1 + (0.137 - 0.238i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (4.27 - 7.40i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (3.63 + 6.30i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 4.27T + 47T^{2} \) |
| 53 | \( 1 - 1.72T + 53T^{2} \) |
| 59 | \( 1 + (7.27 + 12.6i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.63 - 2.83i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.63 + 4.56i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-1 - 1.73i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 - 11.8T + 73T^{2} \) |
| 79 | \( 1 + 0.725T + 79T^{2} \) |
| 83 | \( 1 - 17.0T + 83T^{2} \) |
| 89 | \( 1 + (-2.86 + 4.95i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (3.91 + 6.77i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.235184059604811713806956552723, −8.303622688866384038051745487151, −7.83143918146693904888377338284, −6.69675799879268460658796123634, −6.05842059470533065658400473252, −5.36243694993734310049620528198, −3.83183166030962414472027756745, −3.28254181149445120485908300953, −1.92670211967839181431227630220, −0.77621079495670323449665891843,
1.56494678241911347016437634997, 2.58451573962371584225282479067, 3.76069167920619765663690776411, 4.56764184793435272466297460973, 5.44380021352831010331254476975, 6.52234710593637300417237023694, 7.04124640380508050775812847137, 8.254342228474161716755029859080, 9.146790367494322776299970574576, 9.334895945440894987003511018368