L(s) = 1 | + 1.48i·2-s + (−0.527 + 0.527i)3-s − 0.209·4-s + (0.786 − 0.786i)5-s + (−0.783 − 0.783i)6-s + (0.754 + 0.754i)7-s + 2.66i·8-s + 2.44i·9-s + (1.16 + 1.16i)10-s + (0.707 + 0.707i)11-s + (0.110 − 0.110i)12-s − 1.70·13-s + (−1.12 + 1.12i)14-s + 0.828i·15-s − 4.37·16-s + (−1.83 − 3.69i)17-s + ⋯ |
L(s) = 1 | + 1.05i·2-s + (−0.304 + 0.304i)3-s − 0.104·4-s + (0.351 − 0.351i)5-s + (−0.319 − 0.319i)6-s + (0.284 + 0.284i)7-s + 0.940i·8-s + 0.814i·9-s + (0.369 + 0.369i)10-s + (0.213 + 0.213i)11-s + (0.0319 − 0.0319i)12-s − 0.472·13-s + (−0.299 + 0.299i)14-s + 0.214i·15-s − 1.09·16-s + (−0.444 − 0.895i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.331 - 0.943i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.331 - 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.711774 + 1.00485i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.711774 + 1.00485i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 + (-0.707 - 0.707i)T \) |
| 17 | \( 1 + (1.83 + 3.69i)T \) |
good | 2 | \( 1 - 1.48iT - 2T^{2} \) |
| 3 | \( 1 + (0.527 - 0.527i)T - 3iT^{2} \) |
| 5 | \( 1 + (-0.786 + 0.786i)T - 5iT^{2} \) |
| 7 | \( 1 + (-0.754 - 0.754i)T + 7iT^{2} \) |
| 13 | \( 1 + 1.70T + 13T^{2} \) |
| 19 | \( 1 + 7.65iT - 19T^{2} \) |
| 23 | \( 1 + (-5.73 - 5.73i)T + 23iT^{2} \) |
| 29 | \( 1 + (-0.352 + 0.352i)T - 29iT^{2} \) |
| 31 | \( 1 + (-6.88 + 6.88i)T - 31iT^{2} \) |
| 37 | \( 1 + (2.07 - 2.07i)T - 37iT^{2} \) |
| 41 | \( 1 + (-3.10 - 3.10i)T + 41iT^{2} \) |
| 43 | \( 1 + 12.7iT - 43T^{2} \) |
| 47 | \( 1 + 7.40T + 47T^{2} \) |
| 53 | \( 1 + 4.02iT - 53T^{2} \) |
| 59 | \( 1 - 2.32iT - 59T^{2} \) |
| 61 | \( 1 + (1.68 + 1.68i)T + 61iT^{2} \) |
| 67 | \( 1 - 10.3T + 67T^{2} \) |
| 71 | \( 1 + (-6.62 + 6.62i)T - 71iT^{2} \) |
| 73 | \( 1 + (2.08 - 2.08i)T - 73iT^{2} \) |
| 79 | \( 1 + (9.70 + 9.70i)T + 79iT^{2} \) |
| 83 | \( 1 - 11.9iT - 83T^{2} \) |
| 89 | \( 1 + 0.543T + 89T^{2} \) |
| 97 | \( 1 + (10.0 - 10.0i)T - 97iT^{2} \) |
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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
−13.25979861736563117201320191766, −11.59711971001578563035588453738, −11.17605697624021283931937040215, −9.676997034549089089486522374073, −8.768032696156131352201922370601, −7.56598799106500826630578285068, −6.74323538380585475182756357479, −5.27290407166178191943386955791, −4.90380866049691335247199457068, −2.39915718409127306259557025277,
1.36373669117259041941212931878, 2.94934408834039148748825431713, 4.27422045112317304824318777347, 6.15780419943381494483703962921, 6.85156975323274468964596111333, 8.373934295496365625508893991969, 9.720683720874337568534218927085, 10.49752117207723934827527101861, 11.27439318438696160356680149148, 12.40365220137631395138556278869