L(s) = 1 | + (−2.37 + 2.37i)5-s − 3.64i·7-s + (0.841 − 0.841i)11-s + (2.64 + 2.64i)13-s − 3.06·17-s + (1.64 + 1.64i)19-s + 7.82i·23-s − 6.29i·25-s + (−0.692 − 0.692i)29-s + 0.354·31-s + (8.66 + 8.66i)35-s + (−4.64 + 4.64i)37-s + 6.43i·41-s + (−5.64 + 5.64i)43-s + 11.1·47-s + ⋯ |
L(s) = 1 | + (−1.06 + 1.06i)5-s − 1.37i·7-s + (0.253 − 0.253i)11-s + (0.733 + 0.733i)13-s − 0.744·17-s + (0.377 + 0.377i)19-s + 1.63i·23-s − 1.25i·25-s + (−0.128 − 0.128i)29-s + 0.0636·31-s + (1.46 + 1.46i)35-s + (−0.763 + 0.763i)37-s + 1.00i·41-s + (−0.860 + 0.860i)43-s + 1.63·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.179 - 0.983i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.179 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9445646458\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9445646458\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (2.37 - 2.37i)T - 5iT^{2} \) |
| 7 | \( 1 + 3.64iT - 7T^{2} \) |
| 11 | \( 1 + (-0.841 + 0.841i)T - 11iT^{2} \) |
| 13 | \( 1 + (-2.64 - 2.64i)T + 13iT^{2} \) |
| 17 | \( 1 + 3.06T + 17T^{2} \) |
| 19 | \( 1 + (-1.64 - 1.64i)T + 19iT^{2} \) |
| 23 | \( 1 - 7.82iT - 23T^{2} \) |
| 29 | \( 1 + (0.692 + 0.692i)T + 29iT^{2} \) |
| 31 | \( 1 - 0.354T + 31T^{2} \) |
| 37 | \( 1 + (4.64 - 4.64i)T - 37iT^{2} \) |
| 41 | \( 1 - 6.43iT - 41T^{2} \) |
| 43 | \( 1 + (5.64 - 5.64i)T - 43iT^{2} \) |
| 47 | \( 1 - 11.1T + 47T^{2} \) |
| 53 | \( 1 + (5.44 - 5.44i)T - 53iT^{2} \) |
| 59 | \( 1 + (7.82 - 7.82i)T - 59iT^{2} \) |
| 61 | \( 1 + (4.64 + 4.64i)T + 61iT^{2} \) |
| 67 | \( 1 + (-4 - 4i)T + 67iT^{2} \) |
| 71 | \( 1 + 3.36iT - 71T^{2} \) |
| 73 | \( 1 - 7.29iT - 73T^{2} \) |
| 79 | \( 1 + 4.35T + 79T^{2} \) |
| 83 | \( 1 + (0.841 + 0.841i)T + 83iT^{2} \) |
| 89 | \( 1 - 9.50iT - 89T^{2} \) |
| 97 | \( 1 - 10.5T + 97T^{2} \) |
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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
−10.18566538930561309111306824217, −9.262948923723479998368817506117, −8.163236943438954154289150738258, −7.43601419176944091872095141678, −6.90192428202575893026187134173, −6.07596098853449678128828802293, −4.52876140594663407733657513425, −3.80037909360911735469316046725, −3.16628915083695519403341980304, −1.36294851104108039864558930082,
0.44289040598309111264361564255, 2.09676202345228768698943837084, 3.37727738513199372264794212858, 4.43748477105124474992810939365, 5.19875013876710223584512837521, 6.08191093077905141700127476123, 7.17238716313302499775431403396, 8.227137332602645621858869728160, 8.746503492121296222050612377036, 9.150962929410638613065716274008