L(s) = 1 | + (0.866 + 0.5i)2-s + (−0.130 − 0.991i)3-s + (0.499 + 0.866i)4-s + (0.382 − 0.923i)6-s + 0.999i·8-s + (−0.965 + 0.258i)9-s + (1.22 − 0.707i)11-s + (0.793 − 0.608i)12-s + (−0.5 + 0.866i)16-s + (−0.382 − 0.662i)17-s + (−0.965 − 0.258i)18-s + (1.60 + 0.923i)19-s + 1.41·22-s + (0.991 − 0.130i)24-s + (−0.5 − 0.866i)25-s + ⋯ |
L(s) = 1 | + (0.866 + 0.5i)2-s + (−0.130 − 0.991i)3-s + (0.499 + 0.866i)4-s + (0.382 − 0.923i)6-s + 0.999i·8-s + (−0.965 + 0.258i)9-s + (1.22 − 0.707i)11-s + (0.793 − 0.608i)12-s + (−0.5 + 0.866i)16-s + (−0.382 − 0.662i)17-s + (−0.965 − 0.258i)18-s + (1.60 + 0.923i)19-s + 1.41·22-s + (0.991 − 0.130i)24-s + (−0.5 − 0.866i)25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0350i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0350i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.686193029\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.686193029\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 - 0.5i)T \) |
| 3 | \( 1 + (0.130 + 0.991i)T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( 1 + (0.382 + 0.662i)T + (-0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (-1.60 - 0.923i)T + (0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 + 1.84T + T^{2} \) |
| 43 | \( 1 + 1.41T + T^{2} \) |
| 47 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + (-0.382 - 0.662i)T + (-0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + (0.662 - 0.382i)T + (0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + 0.765T + T^{2} \) |
| 89 | \( 1 + (0.923 - 1.60i)T + (-0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + 1.84iT - T^{2} \) |
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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
−9.954082398750587821962660506114, −8.782325788824092655306664196915, −8.163480685754652888161463359432, −7.24969878495946062107565334848, −6.61811664447869132427178126872, −5.87228854081291024486155960113, −5.08587995431325615091189320310, −3.78982328642598243145098954673, −2.93114561150627256665044767706, −1.55386461065253944314109226859,
1.65016294515927629137116902237, 3.12988592206788613255999555030, 3.83370808906513824137236758346, 4.74290011258802160675799765390, 5.40884084680106131933430881579, 6.43933980993900553381096554542, 7.19012856129294059543652793913, 8.649478948575938813922899370595, 9.542961087132898881569454332658, 9.911870825035617164522026569326