| L(s) = 1 | + (−1.41 − 0.0633i)2-s + (−0.0286 + 0.190i)3-s + (1.99 + 0.179i)4-s + (−3.07 + 1.20i)5-s + (0.0525 − 0.266i)6-s + (0.384 + 2.61i)7-s + (−2.80 − 0.379i)8-s + (2.83 + 0.873i)9-s + (4.41 − 1.50i)10-s + (−0.799 − 2.59i)11-s + (−0.0910 + 0.373i)12-s + (−2.39 − 0.546i)13-s + (−0.376 − 3.72i)14-s + (−0.141 − 0.618i)15-s + (3.93 + 0.713i)16-s + (−4.97 − 3.39i)17-s + ⋯ |
| L(s) = 1 | + (−0.998 − 0.0448i)2-s + (−0.0165 + 0.109i)3-s + (0.995 + 0.0895i)4-s + (−1.37 + 0.538i)5-s + (0.0214 − 0.108i)6-s + (0.145 + 0.989i)7-s + (−0.990 − 0.134i)8-s + (0.943 + 0.291i)9-s + (1.39 − 0.476i)10-s + (−0.241 − 0.781i)11-s + (−0.0262 + 0.107i)12-s + (−0.663 − 0.151i)13-s + (−0.100 − 0.994i)14-s + (−0.0364 − 0.159i)15-s + (0.983 + 0.178i)16-s + (−1.20 − 0.823i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.993 + 0.113i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.993 + 0.113i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.00755680 - 0.132315i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00755680 - 0.132315i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.41 + 0.0633i)T \) |
| 7 | \( 1 + (-0.384 - 2.61i)T \) |
| good | 3 | \( 1 + (0.0286 - 0.190i)T + (-2.86 - 0.884i)T^{2} \) |
| 5 | \( 1 + (3.07 - 1.20i)T + (3.66 - 3.40i)T^{2} \) |
| 11 | \( 1 + (0.799 + 2.59i)T + (-9.08 + 6.19i)T^{2} \) |
| 13 | \( 1 + (2.39 + 0.546i)T + (11.7 + 5.64i)T^{2} \) |
| 17 | \( 1 + (4.97 + 3.39i)T + (6.21 + 15.8i)T^{2} \) |
| 19 | \( 1 + (2.12 + 1.22i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (4.24 - 2.89i)T + (8.40 - 21.4i)T^{2} \) |
| 29 | \( 1 + (-2.12 - 4.40i)T + (-18.0 + 22.6i)T^{2} \) |
| 31 | \( 1 + (2.86 + 4.95i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (3.51 + 0.263i)T + (36.5 + 5.51i)T^{2} \) |
| 41 | \( 1 + (1.26 - 1.58i)T + (-9.12 - 39.9i)T^{2} \) |
| 43 | \( 1 + (8.29 - 6.61i)T + (9.56 - 41.9i)T^{2} \) |
| 47 | \( 1 + (4.91 + 4.56i)T + (3.51 + 46.8i)T^{2} \) |
| 53 | \( 1 + (-11.7 + 0.883i)T + (52.4 - 7.89i)T^{2} \) |
| 59 | \( 1 + (8.66 + 3.40i)T + (43.2 + 40.1i)T^{2} \) |
| 61 | \( 1 + (-2.16 - 0.162i)T + (60.3 + 9.09i)T^{2} \) |
| 67 | \( 1 + (-4.07 + 2.35i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (7.32 + 3.52i)T + (44.2 + 55.5i)T^{2} \) |
| 73 | \( 1 + (-1.71 + 1.58i)T + (5.45 - 72.7i)T^{2} \) |
| 79 | \( 1 + (4.47 - 7.75i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-3.55 + 0.812i)T + (74.7 - 36.0i)T^{2} \) |
| 89 | \( 1 + (-14.8 - 4.56i)T + (73.5 + 50.1i)T^{2} \) |
| 97 | \( 1 - 8.27T + 97T^{2} \) |
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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
−11.53520122432092534066203892830, −10.95358685542608037462955273856, −9.958077126549933631746991169153, −8.929091448647408839023845972393, −8.101093071779012820550489645104, −7.35144056831908495944924178991, −6.47041469876579084528526657759, −4.95106107860736395798777064676, −3.48666331579364308547537127326, −2.26351162258688447977150298035,
0.11500522851580693043200069656, 1.81582601391938178340592869829, 3.87590583085935309916811160123, 4.61652630873436113864370098876, 6.61906635684597919445260015949, 7.27033248626776224576710721449, 8.017234274412117130950106864089, 8.822060603165199936113709501965, 10.11900463132020567020983310730, 10.53943528364549578751242382234
