| L(s) = 1 | + (0.654 + 0.755i)2-s + (−1.71 − 1.10i)3-s + (−0.142 + 0.989i)4-s + (0.415 − 0.909i)5-s + (−0.290 − 2.01i)6-s + (3.10 + 0.911i)7-s + (−0.841 + 0.540i)8-s + (0.481 + 1.05i)9-s + (0.959 − 0.281i)10-s + (1.85 − 2.13i)11-s + (1.33 − 1.54i)12-s + (5.91 − 1.73i)13-s + (1.34 + 2.94i)14-s + (−1.71 + 1.10i)15-s + (−0.959 − 0.281i)16-s + (−0.475 − 3.30i)17-s + ⋯ |
| L(s) = 1 | + (0.463 + 0.534i)2-s + (−0.990 − 0.636i)3-s + (−0.0711 + 0.494i)4-s + (0.185 − 0.406i)5-s + (−0.118 − 0.824i)6-s + (1.17 + 0.344i)7-s + (−0.297 + 0.191i)8-s + (0.160 + 0.351i)9-s + (0.303 − 0.0890i)10-s + (0.558 − 0.644i)11-s + (0.385 − 0.444i)12-s + (1.64 − 0.481i)13-s + (0.359 + 0.786i)14-s + (−0.442 + 0.284i)15-s + (−0.239 − 0.0704i)16-s + (−0.115 − 0.802i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0655i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0655i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.32704 - 0.0435110i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.32704 - 0.0435110i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.654 - 0.755i)T \) |
| 5 | \( 1 + (-0.415 + 0.909i)T \) |
| 23 | \( 1 + (0.991 + 4.69i)T \) |
| good | 3 | \( 1 + (1.71 + 1.10i)T + (1.24 + 2.72i)T^{2} \) |
| 7 | \( 1 + (-3.10 - 0.911i)T + (5.88 + 3.78i)T^{2} \) |
| 11 | \( 1 + (-1.85 + 2.13i)T + (-1.56 - 10.8i)T^{2} \) |
| 13 | \( 1 + (-5.91 + 1.73i)T + (10.9 - 7.02i)T^{2} \) |
| 17 | \( 1 + (0.475 + 3.30i)T + (-16.3 + 4.78i)T^{2} \) |
| 19 | \( 1 + (1.06 - 7.38i)T + (-18.2 - 5.35i)T^{2} \) |
| 29 | \( 1 + (-0.656 - 4.56i)T + (-27.8 + 8.17i)T^{2} \) |
| 31 | \( 1 + (6.36 - 4.09i)T + (12.8 - 28.1i)T^{2} \) |
| 37 | \( 1 + (4.04 + 8.85i)T + (-24.2 + 27.9i)T^{2} \) |
| 41 | \( 1 + (0.910 - 1.99i)T + (-26.8 - 30.9i)T^{2} \) |
| 43 | \( 1 + (-0.0224 - 0.0143i)T + (17.8 + 39.1i)T^{2} \) |
| 47 | \( 1 + 7.99T + 47T^{2} \) |
| 53 | \( 1 + (-10.5 - 3.11i)T + (44.5 + 28.6i)T^{2} \) |
| 59 | \( 1 + (-0.649 + 0.190i)T + (49.6 - 31.8i)T^{2} \) |
| 61 | \( 1 + (-1.17 + 0.758i)T + (25.3 - 55.4i)T^{2} \) |
| 67 | \( 1 + (-6.56 - 7.58i)T + (-9.53 + 66.3i)T^{2} \) |
| 71 | \( 1 + (-0.633 - 0.731i)T + (-10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (0.492 - 3.42i)T + (-70.0 - 20.5i)T^{2} \) |
| 79 | \( 1 + (11.6 - 3.42i)T + (66.4 - 42.7i)T^{2} \) |
| 83 | \( 1 + (-0.439 - 0.961i)T + (-54.3 + 62.7i)T^{2} \) |
| 89 | \( 1 + (8.63 + 5.54i)T + (36.9 + 80.9i)T^{2} \) |
| 97 | \( 1 + (6.28 - 13.7i)T + (-63.5 - 73.3i)T^{2} \) |
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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
−12.25391498580551229677327836855, −11.44194851609006621595733197100, −10.70995592747883228254920030096, −8.816927598604100755148824540016, −8.236393917274287616720081355468, −6.88424698881526038073798204025, −5.84582571698687551774745628255, −5.33680429679015797486282685934, −3.79670039925856555003677108077, −1.39229113688860041589246136403,
1.72788168464991073600816936026, 3.88249727204966637849229621630, 4.71224787689084524347023239264, 5.80548706419221110646068813535, 6.84078392156420684882790507407, 8.422971700039378422447506873735, 9.672720233815281602722970896928, 10.72398277016165640838129504924, 11.29000175557941437193473810770, 11.68830900918857219667622056101
