L(s) = 1 | + (−1.86 + 0.604i)2-s + (−0.506 + 0.506i)3-s + (2.28 − 1.66i)4-s + (0.636 − 1.24i)6-s + (−2.10 + 2.89i)8-s + 0.486i·9-s + (−0.316 + 2.00i)12-s + (−1.38 − 0.705i)13-s + (1.28 − 3.96i)16-s + (−0.293 − 0.904i)18-s + (0.309 + 0.951i)23-s + (−0.400 − 2.53i)24-s + (−0.309 + 0.951i)25-s + (3.00 + 0.475i)26-s + (−0.753 − 0.753i)27-s + ⋯ |
L(s) = 1 | + (−1.86 + 0.604i)2-s + (−0.506 + 0.506i)3-s + (2.28 − 1.66i)4-s + (0.636 − 1.24i)6-s + (−2.10 + 2.89i)8-s + 0.486i·9-s + (−0.316 + 2.00i)12-s + (−1.38 − 0.705i)13-s + (1.28 − 3.96i)16-s + (−0.293 − 0.904i)18-s + (0.309 + 0.951i)23-s + (−0.400 − 2.53i)24-s + (−0.309 + 0.951i)25-s + (3.00 + 0.475i)26-s + (−0.753 − 0.753i)27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 943 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.975 - 0.219i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 943 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.975 - 0.219i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2125741432\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2125741432\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 23 | \( 1 + (-0.309 - 0.951i)T \) |
| 41 | \( 1 + (0.978 + 0.207i)T \) |
good | 2 | \( 1 + (1.86 - 0.604i)T + (0.809 - 0.587i)T^{2} \) |
| 3 | \( 1 + (0.506 - 0.506i)T - iT^{2} \) |
| 5 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 7 | \( 1 + (0.587 - 0.809i)T^{2} \) |
| 11 | \( 1 + (-0.951 + 0.309i)T^{2} \) |
| 13 | \( 1 + (1.38 + 0.705i)T + (0.587 + 0.809i)T^{2} \) |
| 17 | \( 1 + (0.951 - 0.309i)T^{2} \) |
| 19 | \( 1 + (-0.587 + 0.809i)T^{2} \) |
| 29 | \( 1 + (0.292 - 1.84i)T + (-0.951 - 0.309i)T^{2} \) |
| 31 | \( 1 + (-0.658 - 0.478i)T + (0.309 + 0.951i)T^{2} \) |
| 37 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 43 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 47 | \( 1 + (-0.0475 + 0.0932i)T + (-0.587 - 0.809i)T^{2} \) |
| 53 | \( 1 + (0.951 + 0.309i)T^{2} \) |
| 59 | \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \) |
| 61 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 67 | \( 1 + (-0.951 - 0.309i)T^{2} \) |
| 71 | \( 1 + (1.07 - 0.170i)T + (0.951 - 0.309i)T^{2} \) |
| 73 | \( 1 - 1.98iT - T^{2} \) |
| 79 | \( 1 + iT^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + (0.587 - 0.809i)T^{2} \) |
| 97 | \( 1 + (-0.951 - 0.309i)T^{2} \) |
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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
−10.35539562738320533368689477586, −9.789870288774525057264829946646, −9.096879961118366754791261926807, −8.101932193396128908316870091106, −7.45141547061495282814828672541, −6.75942282616919671230805687003, −5.48838649286637172713939797654, −5.10542799970754128133918807136, −2.96173965335124078432785729464, −1.62088859145956747480788306046,
0.38354315758073013140926621289, 1.88887943844739407567196683642, 2.82008836860571423314636731758, 4.29287543657237716129239810484, 6.11825804292176088501691098065, 6.75414714454596866419071650628, 7.51108860576183491973801548912, 8.278111183812528481363683314347, 9.212060366901313936482954605284, 9.826493779479039967893125099281