L(s) = 1 | + (−0.5 + 1.53i)2-s + (−0.5 + 0.363i)3-s + (−1.30 − 0.951i)4-s + (0.309 + 0.951i)5-s + (−0.309 − 0.951i)6-s + (0.809 − 0.587i)8-s + (−0.190 + 0.587i)9-s − 1.61·10-s + (0.309 − 0.951i)11-s + 12-s + (−0.5 + 1.53i)13-s + (−0.5 − 0.363i)15-s + (−0.809 − 0.587i)18-s + (−0.809 + 0.587i)19-s + (0.499 − 1.53i)20-s + ⋯ |
L(s) = 1 | + (−0.5 + 1.53i)2-s + (−0.5 + 0.363i)3-s + (−1.30 − 0.951i)4-s + (0.309 + 0.951i)5-s + (−0.309 − 0.951i)6-s + (0.809 − 0.587i)8-s + (−0.190 + 0.587i)9-s − 1.61·10-s + (0.309 − 0.951i)11-s + 12-s + (−0.5 + 1.53i)13-s + (−0.5 − 0.363i)15-s + (−0.809 − 0.587i)18-s + (−0.809 + 0.587i)19-s + (0.499 − 1.53i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.550 + 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.550 + 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4965845369\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4965845369\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (-0.309 - 0.951i)T \) |
| 11 | \( 1 + (-0.309 + 0.951i)T \) |
| 19 | \( 1 + (0.809 - 0.587i)T \) |
good | 2 | \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \) |
| 3 | \( 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2} \) |
| 7 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 13 | \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \) |
| 17 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 31 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 37 | \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \) |
| 41 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 53 | \( 1 + (-0.618 + 1.90i)T + (-0.809 - 0.587i)T^{2} \) |
| 59 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 61 | \( 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2} \) |
| 67 | \( 1 - 0.618T + T^{2} \) |
| 71 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 73 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 79 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 83 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \) |
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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
−10.46928681669208622789042408856, −9.674295193824435293843026368980, −8.871427741003624023628505384389, −8.076890547637702730937032525175, −7.15355582849844669644080438137, −6.47137403209915347191360585446, −5.88313498391416528075764014017, −5.00526866480211695744943522114, −3.89082791983339165068547208944, −2.26130537906733152861242861994,
0.57402276687969169064111777264, 1.74387819764198111530161953332, 2.86948013701621776684247653956, 4.08275180719986615025978957516, 5.04628685002785603161654090759, 6.03516447731750310650178453806, 7.18013880145692810304374731024, 8.336973893399868681911700609163, 8.993411325280886047778675526979, 9.781945118835883919666537851535