L(s) = 1 | + (0.564 + 1.73i)2-s + (−0.978 + 0.207i)3-s + (−1.89 + 1.37i)4-s + (−0.5 + 0.866i)5-s + (−0.913 − 1.58i)6-s + (−1.97 − 1.43i)8-s + (0.913 − 0.406i)9-s + (−1.78 − 0.379i)10-s + (1.56 − 1.73i)12-s + (0.309 − 0.951i)15-s + (0.657 − 2.02i)16-s + (0.104 − 0.994i)17-s + (1.22 + 1.35i)18-s + (−1.30 + 1.45i)19-s + (−0.244 − 2.32i)20-s + ⋯ |
L(s) = 1 | + (0.564 + 1.73i)2-s + (−0.978 + 0.207i)3-s + (−1.89 + 1.37i)4-s + (−0.5 + 0.866i)5-s + (−0.913 − 1.58i)6-s + (−1.97 − 1.43i)8-s + (0.913 − 0.406i)9-s + (−1.78 − 0.379i)10-s + (1.56 − 1.73i)12-s + (0.309 − 0.951i)15-s + (0.657 − 2.02i)16-s + (0.104 − 0.994i)17-s + (1.22 + 1.35i)18-s + (−1.30 + 1.45i)19-s + (−0.244 − 2.32i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.807 + 0.589i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.807 + 0.589i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6341685065\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6341685065\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.978 - 0.207i)T \) |
| 5 | \( 1 + (0.5 - 0.866i)T \) |
| 31 | \( 1 + (-0.309 - 0.951i)T \) |
good | 2 | \( 1 + (-0.564 - 1.73i)T + (-0.809 + 0.587i)T^{2} \) |
| 7 | \( 1 + (-0.669 - 0.743i)T^{2} \) |
| 11 | \( 1 + (0.978 - 0.207i)T^{2} \) |
| 13 | \( 1 + (0.104 - 0.994i)T^{2} \) |
| 17 | \( 1 + (-0.104 + 0.994i)T + (-0.978 - 0.207i)T^{2} \) |
| 19 | \( 1 + (1.30 - 1.45i)T + (-0.104 - 0.994i)T^{2} \) |
| 23 | \( 1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2} \) |
| 29 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 37 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 + (-0.913 - 0.406i)T^{2} \) |
| 43 | \( 1 + (0.104 + 0.994i)T^{2} \) |
| 47 | \( 1 + (0.0646 - 0.198i)T + (-0.809 - 0.587i)T^{2} \) |
| 53 | \( 1 + (-0.564 + 0.251i)T + (0.669 - 0.743i)T^{2} \) |
| 59 | \( 1 + (-0.913 + 0.406i)T^{2} \) |
| 61 | \( 1 - 1.33T + T^{2} \) |
| 67 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + (-0.669 + 0.743i)T^{2} \) |
| 73 | \( 1 + (0.978 - 0.207i)T^{2} \) |
| 79 | \( 1 + (-0.204 + 1.94i)T + (-0.978 - 0.207i)T^{2} \) |
| 83 | \( 1 + (-0.204 - 0.0434i)T + (0.913 + 0.406i)T^{2} \) |
| 89 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 97 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
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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
−11.95250851977365815982932783258, −10.95762606434980731028558904527, −9.983555048463762083558227390202, −8.795820321948485821779902026666, −7.66910044589881115739464208820, −7.01406085976307191397190052363, −6.28857086626526214160568840154, −5.41806519964115899310731565181, −4.45186882938014062509276643625, −3.47246675472654939217306315104,
0.819544924012247194107594640238, 2.28910620691738664547231703500, 3.98092773121881294086306835642, 4.64018499297231635875271492843, 5.48210693948639976931339796895, 6.76581536877239251743505313699, 8.369928759899555932128945517606, 9.216420717740599860379852416222, 10.32605622475222292906673084854, 10.98237512199845388540162718731