| L(s) = 1 | + (1.01 + 0.215i)2-s + (0.967 + 1.67i)3-s + (−0.845 − 0.376i)4-s + (0.246 + 2.34i)5-s + (0.619 + 1.90i)6-s + (2.64 + 0.0146i)7-s + (−2.45 − 1.78i)8-s + (−0.371 + 0.643i)9-s + (−0.255 + 2.42i)10-s + (−0.644 + 6.13i)11-s + (−0.187 − 1.78i)12-s + (−2.18 − 6.71i)13-s + (2.67 + 0.584i)14-s + (−3.68 + 2.67i)15-s + (−0.863 − 0.959i)16-s + (−0.0926 + 0.881i)17-s + ⋯ |
| L(s) = 1 | + (0.716 + 0.152i)2-s + (0.558 + 0.967i)3-s + (−0.422 − 0.188i)4-s + (0.110 + 1.04i)5-s + (0.252 + 0.778i)6-s + (0.999 + 0.00553i)7-s + (−0.867 − 0.630i)8-s + (−0.123 + 0.214i)9-s + (−0.0807 + 0.768i)10-s + (−0.194 + 1.85i)11-s + (−0.0540 − 0.514i)12-s + (−0.604 − 1.86i)13-s + (0.715 + 0.156i)14-s + (−0.952 + 0.691i)15-s + (−0.215 − 0.239i)16-s + (−0.0224 + 0.213i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.188 - 0.982i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.188 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.52530 + 1.26005i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.52530 + 1.26005i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (-2.64 - 0.0146i)T \) |
| 41 | \( 1 + (-2.79 + 5.76i)T \) |
| good | 2 | \( 1 + (-1.01 - 0.215i)T + (1.82 + 0.813i)T^{2} \) |
| 3 | \( 1 + (-0.967 - 1.67i)T + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (-0.246 - 2.34i)T + (-4.89 + 1.03i)T^{2} \) |
| 11 | \( 1 + (0.644 - 6.13i)T + (-10.7 - 2.28i)T^{2} \) |
| 13 | \( 1 + (2.18 + 6.71i)T + (-10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (0.0926 - 0.881i)T + (-16.6 - 3.53i)T^{2} \) |
| 19 | \( 1 + (-2.08 - 2.32i)T + (-1.98 + 18.8i)T^{2} \) |
| 23 | \( 1 + (0.117 + 0.0250i)T + (21.0 + 9.35i)T^{2} \) |
| 29 | \( 1 + (0.532 - 0.387i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (-1.09 + 10.4i)T + (-30.3 - 6.44i)T^{2} \) |
| 37 | \( 1 + (0.752 + 7.15i)T + (-36.1 + 7.69i)T^{2} \) |
| 43 | \( 1 + (-0.931 - 2.86i)T + (-34.7 + 25.2i)T^{2} \) |
| 47 | \( 1 + (7.00 + 1.48i)T + (42.9 + 19.1i)T^{2} \) |
| 53 | \( 1 + (7.79 + 3.46i)T + (35.4 + 39.3i)T^{2} \) |
| 59 | \( 1 + (1.67 - 1.86i)T + (-6.16 - 58.6i)T^{2} \) |
| 61 | \( 1 + (-3.22 - 3.57i)T + (-6.37 + 60.6i)T^{2} \) |
| 67 | \( 1 + (-0.951 - 0.423i)T + (44.8 + 49.7i)T^{2} \) |
| 71 | \( 1 + (-2.43 - 1.77i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-1.11 - 1.93i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (3.24 - 5.61i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 2.83T + 83T^{2} \) |
| 89 | \( 1 + (5.17 + 5.74i)T + (-9.30 + 88.5i)T^{2} \) |
| 97 | \( 1 + (2.69 - 1.95i)T + (29.9 - 92.2i)T^{2} \) |
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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
−12.26026735958816063344937485248, −10.82269749424291685097544841719, −10.00757404720397728849828044835, −9.588099874595768870873015756161, −8.104801583687640544890708625172, −7.19351552347477706889823818884, −5.65715283863780993309740495471, −4.76020872528101853845868831424, −3.84334426543818314522552443412, −2.60287230772585607648604736605,
1.41036481179283102003889306541, 2.96364185550220101369415375541, 4.56345199615537112794397445801, 5.19167556607772561893029815509, 6.65131005567776648981873690205, 8.027945957740360853793784621284, 8.601316092004867918860744493299, 9.252400669571442434600229362670, 11.17327024507642881327345941692, 11.88048639082460499045953770819
