L(s) = 1 | + (−2.02 − 1.69i)2-s + (1.43 + 0.962i)3-s + (0.865 + 4.90i)4-s + (−0.259 − 0.712i)5-s + (−1.28 − 4.39i)6-s + (−1.63 + 2.82i)7-s + (3.94 − 6.83i)8-s + (1.14 + 2.77i)9-s + (−0.685 + 1.88i)10-s + 6.35i·11-s + (−3.47 + 7.90i)12-s + (0.295 − 0.812i)13-s + (8.09 − 2.94i)14-s + (0.312 − 1.27i)15-s + (−10.2 + 3.71i)16-s + (−0.818 − 2.24i)17-s + ⋯ |
L(s) = 1 | + (−1.43 − 1.20i)2-s + (0.831 + 0.555i)3-s + (0.432 + 2.45i)4-s + (−0.116 − 0.318i)5-s + (−0.522 − 1.79i)6-s + (−0.616 + 1.06i)7-s + (1.39 − 2.41i)8-s + (0.382 + 0.924i)9-s + (−0.216 + 0.595i)10-s + 1.91i·11-s + (−1.00 + 2.28i)12-s + (0.0819 − 0.225i)13-s + (2.16 − 0.787i)14-s + (0.0807 − 0.329i)15-s + (−2.55 + 0.929i)16-s + (−0.198 − 0.545i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.951 - 0.308i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.951 - 0.308i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.672309 + 0.106121i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.672309 + 0.106121i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-1.43 - 0.962i)T \) |
| 19 | \( 1 + (-4.31 + 0.649i)T \) |
good | 2 | \( 1 + (2.02 + 1.69i)T + (0.347 + 1.96i)T^{2} \) |
| 5 | \( 1 + (0.259 + 0.712i)T + (-3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (1.63 - 2.82i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 - 6.35iT - 11T^{2} \) |
| 13 | \( 1 + (-0.295 + 0.812i)T + (-9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (0.818 + 2.24i)T + (-13.0 + 10.9i)T^{2} \) |
| 23 | \( 1 + (-0.245 + 0.0432i)T + (21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (0.803 + 4.55i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 - 1.49iT - 31T^{2} \) |
| 37 | \( 1 - 4.34iT - 37T^{2} \) |
| 41 | \( 1 + (-1.48 - 1.24i)T + (7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-0.639 + 3.62i)T + (-40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (7.25 - 1.27i)T + (44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + (-7.32 + 6.14i)T + (9.20 - 52.1i)T^{2} \) |
| 59 | \( 1 + (0.645 - 3.66i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (5.85 + 2.13i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (1.22 + 1.45i)T + (-11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (0.638 + 0.536i)T + (12.3 + 69.9i)T^{2} \) |
| 73 | \( 1 + (-1.84 + 10.4i)T + (-68.5 - 24.9i)T^{2} \) |
| 79 | \( 1 + (4.52 + 12.4i)T + (-60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (-6.20 - 3.58i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-2.93 - 16.6i)T + (-83.6 + 30.4i)T^{2} \) |
| 97 | \( 1 + (-7.16 + 8.54i)T + (-16.8 - 95.5i)T^{2} \) |
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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
−12.42678682645373707321231573725, −11.78969999420474869544315099414, −10.40087470491476581794239535989, −9.608199331442356168079118185972, −9.194213046151587076831759159213, −8.164208492368561963448489455192, −7.16412005542961406288232511222, −4.67491538701041632795290641872, −3.10862024862813767704617679355, −2.10053304619841469228837333858,
0.978352298959568616332738903234, 3.40828810729641476398559362052, 5.88801781698571107585458883731, 6.82928673836230745894635341707, 7.56367304875898866757948482950, 8.523584756540126612111964193544, 9.275136971840450845772634781174, 10.36513815290048561583319827309, 11.23506981532774528340806568067, 13.17540009117842543337854158639