L(s) = 1 | + (−0.5 + 0.866i)3-s + (0.5 − 0.866i)7-s + (−0.5 − 0.866i)9-s − 11-s + (−0.5 + 0.866i)13-s + (0.5 + 0.866i)17-s + (0.5 − 0.866i)19-s + (0.5 + 0.866i)21-s − 23-s + 27-s − 29-s + 31-s + (0.5 − 0.866i)33-s + (−0.5 − 0.866i)39-s + (−0.5 + 0.866i)41-s + ⋯ |
L(s) = 1 | + (−0.5 + 0.866i)3-s + (0.5 − 0.866i)7-s + (−0.5 − 0.866i)9-s − 11-s + (−0.5 + 0.866i)13-s + (0.5 + 0.866i)17-s + (0.5 − 0.866i)19-s + (0.5 + 0.866i)21-s − 23-s + 27-s − 29-s + 31-s + (0.5 − 0.866i)33-s + (−0.5 − 0.866i)39-s + (−0.5 + 0.866i)41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.989 - 0.146i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.989 - 0.146i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.01568221815 + 0.2128307378i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.01568221815 + 0.2128307378i\) |
\(L(1)\) |
\(\approx\) |
\(0.6944007980 + 0.1649494522i\) |
\(L(1)\) |
\(\approx\) |
\(0.6944007980 + 0.1649494522i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 37 | \( 1 \) |
good | 3 | \( 1 + (-0.5 + 0.866i)T \) |
| 7 | \( 1 + (0.5 - 0.866i)T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + (-0.5 + 0.866i)T \) |
| 17 | \( 1 + (0.5 + 0.866i)T \) |
| 19 | \( 1 + (0.5 - 0.866i)T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| 31 | \( 1 + T \) |
| 41 | \( 1 + (-0.5 + 0.866i)T \) |
| 43 | \( 1 + T \) |
| 47 | \( 1 - T \) |
| 53 | \( 1 + (-0.5 - 0.866i)T \) |
| 59 | \( 1 + (0.5 + 0.866i)T \) |
| 61 | \( 1 + (0.5 - 0.866i)T \) |
| 67 | \( 1 + (-0.5 + 0.866i)T \) |
| 71 | \( 1 + (-0.5 + 0.866i)T \) |
| 73 | \( 1 - T \) |
| 79 | \( 1 + (-0.5 + 0.866i)T \) |
| 83 | \( 1 + (-0.5 - 0.866i)T \) |
| 89 | \( 1 + (-0.5 - 0.866i)T \) |
| 97 | \( 1 - T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−20.43081518602962842274451078572, −19.21848478895536605226692742475, −18.69478658858033488634593076876, −17.975792872497148180874326337562, −17.61489544342606196108661099284, −16.493552356580290894970153138626, −15.84714360746877992193956133204, −14.964998076515176505616279428739, −14.09571107343558561363945628339, −13.38343077647096708347628151730, −12.37732076936191093058883684409, −12.11971694694043093625825331473, −11.21439482556973011312805900516, −10.35988965493714589834121427476, −9.52959782366720566773375829024, −8.21953262699889071388104162460, −7.88961824676115342666958070376, −7.10085608996737396113032970931, −5.71091062764002344508347935505, −5.61927033474742861433503594315, −4.65575645993154591568612733840, −3.084531128333389644863581733642, −2.38681478199383243037582026125, −1.445553195537027892382377211340, −0.08661121336054289709167039628,
1.3313691952670408151916761523, 2.60384722118546512246338213101, 3.70930761751718343635122383465, 4.44665208100295110382532443073, 5.1135635980151494322208381268, 6.00254935723195289793200965126, 6.99687270919111146573122395479, 7.850874218386591581316568683406, 8.69914306057055087545872741353, 9.931210357600576749957765337498, 10.082650222892828228258525917, 11.19571880527002322568726520726, 11.56695709156216126671819487631, 12.65120683524933462446802305403, 13.544954398373188855980504449851, 14.4018085945410598690156608020, 14.998538150484987911594353296015, 15.97563555136990383514480465913, 16.44935159012465869139823991518, 17.35424291559383956656780404392, 17.73997116734827507350913198584, 18.77568623459993283384087160070, 19.71447640531744152189190980985, 20.47411613048498445270851691274, 21.11049619893480765038395749942