L(s) = 1 | + (−0.773 − 0.634i)3-s + (−0.290 − 0.956i)5-s + (0.195 + 0.980i)9-s + (−0.0980 + 0.995i)11-s + (0.956 + 0.290i)13-s + (−0.382 + 0.923i)15-s + (0.382 + 0.923i)17-s + (0.471 + 0.881i)19-s + (0.555 − 0.831i)23-s + (−0.831 + 0.555i)25-s + (0.471 − 0.881i)27-s + (0.995 − 0.0980i)29-s + (0.707 + 0.707i)31-s + (0.707 − 0.707i)33-s + (0.881 + 0.471i)37-s + ⋯ |
L(s) = 1 | + (−0.773 − 0.634i)3-s + (−0.290 − 0.956i)5-s + (0.195 + 0.980i)9-s + (−0.0980 + 0.995i)11-s + (0.956 + 0.290i)13-s + (−0.382 + 0.923i)15-s + (0.382 + 0.923i)17-s + (0.471 + 0.881i)19-s + (0.555 − 0.831i)23-s + (−0.831 + 0.555i)25-s + (0.471 − 0.881i)27-s + (0.995 − 0.0980i)29-s + (0.707 + 0.707i)31-s + (0.707 − 0.707i)33-s + (0.881 + 0.471i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0245 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0245 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7653101428 + 0.7843280188i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7653101428 + 0.7843280188i\) |
\(L(1)\) |
\(\approx\) |
\(0.8325174981 - 0.05687068081i\) |
\(L(1)\) |
\(\approx\) |
\(0.8325174981 - 0.05687068081i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (-0.773 - 0.634i)T \) |
| 5 | \( 1 + (-0.290 - 0.956i)T \) |
| 11 | \( 1 + (-0.0980 + 0.995i)T \) |
| 13 | \( 1 + (0.956 + 0.290i)T \) |
| 17 | \( 1 + (0.382 + 0.923i)T \) |
| 19 | \( 1 + (0.471 + 0.881i)T \) |
| 23 | \( 1 + (0.555 - 0.831i)T \) |
| 29 | \( 1 + (0.995 - 0.0980i)T \) |
| 31 | \( 1 + (0.707 + 0.707i)T \) |
| 37 | \( 1 + (0.881 + 0.471i)T \) |
| 41 | \( 1 + (0.831 + 0.555i)T \) |
| 43 | \( 1 + (-0.773 + 0.634i)T \) |
| 47 | \( 1 + (-0.923 + 0.382i)T \) |
| 53 | \( 1 + (0.995 + 0.0980i)T \) |
| 59 | \( 1 + (0.956 - 0.290i)T \) |
| 61 | \( 1 + (-0.634 + 0.773i)T \) |
| 67 | \( 1 + (-0.634 + 0.773i)T \) |
| 71 | \( 1 + (-0.195 + 0.980i)T \) |
| 73 | \( 1 + (-0.980 + 0.195i)T \) |
| 79 | \( 1 + (-0.923 - 0.382i)T \) |
| 83 | \( 1 + (-0.881 + 0.471i)T \) |
| 89 | \( 1 + (-0.555 - 0.831i)T \) |
| 97 | \( 1 + (-0.707 - 0.707i)T \) |
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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
−19.776158503103030539552577376712, −19.00168493510655298735600041331, −18.15699207128492698422079447250, −17.84936565302606528493128854548, −16.77792541297081271204443202323, −16.019395368042674332187900623755, −15.58517769563827124824353755329, −14.84053832056931542294195103844, −13.84796753355556564712063108457, −13.317736102478633600686629284100, −12.002906231904937983152754130973, −11.3919604602746342989624420332, −10.98978066675150346590174357495, −10.18795114122406159054765848021, −9.403668952063732884941506491464, −8.501396903267414051752125035365, −7.47436581335585561426302366282, −6.67996840472034566795352726960, −5.92337117425522752964490169321, −5.24214766866903646604836511805, −4.18720393626133311590976012204, −3.317634667418018386166198455797, −2.78692835669702517273145707377, −1.04058716856717365541067123272, −0.27316406319554944509497504636,
1.21574575881443813207411507728, 1.35624209003864699518659529282, 2.75488792114239088567796109829, 4.15179135357925432150080800326, 4.66377843984236560612484683719, 5.62826067140126451845608550557, 6.31858158366795978165379308514, 7.178933986685109494863716530015, 8.14368100697395213295987197359, 8.52736940929617157145549056381, 9.81633226753145238648411256679, 10.42159185368596140294052544245, 11.49421102293141225589099514310, 12.007285064668003893084974927941, 12.80554893623365540248760165949, 13.133352269436716123169584307, 14.20021877489819259938120519599, 15.102925063739708918513573263703, 16.190911377177198705958026974085, 16.39703125415422386994246583580, 17.33574699917595144254369026381, 17.91384085049548771260778793255, 18.68143719628905594768717372024, 19.44304125837166232078559602735, 20.13912164518481089495963820319