| L(s) = 1 | + (0.654 + 0.755i)3-s + (0.373 + 2.59i)5-s + (−0.503 + 1.10i)7-s + (−0.142 + 0.989i)9-s + (4.11 + 1.20i)11-s + (−0.475 − 1.04i)13-s + (−1.71 + 1.98i)15-s + (−5.64 − 3.62i)17-s + (−3.45 + 2.21i)19-s + (−1.16 + 0.341i)21-s + (1.47 + 4.56i)23-s + (−1.81 + 0.533i)25-s + (−0.841 + 0.540i)27-s + (6.89 + 4.43i)29-s + (−4.60 + 5.31i)31-s + ⋯ |
| L(s) = 1 | + (0.378 + 0.436i)3-s + (0.167 + 1.16i)5-s + (−0.190 + 0.416i)7-s + (−0.0474 + 0.329i)9-s + (1.23 + 0.364i)11-s + (−0.132 − 0.289i)13-s + (−0.443 + 0.512i)15-s + (−1.36 − 0.879i)17-s + (−0.791 + 0.508i)19-s + (−0.253 + 0.0745i)21-s + (0.307 + 0.951i)23-s + (−0.363 + 0.106i)25-s + (−0.161 + 0.104i)27-s + (1.28 + 0.823i)29-s + (−0.827 + 0.955i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.146 - 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.146 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.03804 + 1.20322i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.03804 + 1.20322i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.654 - 0.755i)T \) |
| 23 | \( 1 + (-1.47 - 4.56i)T \) |
| good | 5 | \( 1 + (-0.373 - 2.59i)T + (-4.79 + 1.40i)T^{2} \) |
| 7 | \( 1 + (0.503 - 1.10i)T + (-4.58 - 5.29i)T^{2} \) |
| 11 | \( 1 + (-4.11 - 1.20i)T + (9.25 + 5.94i)T^{2} \) |
| 13 | \( 1 + (0.475 + 1.04i)T + (-8.51 + 9.82i)T^{2} \) |
| 17 | \( 1 + (5.64 + 3.62i)T + (7.06 + 15.4i)T^{2} \) |
| 19 | \( 1 + (3.45 - 2.21i)T + (7.89 - 17.2i)T^{2} \) |
| 29 | \( 1 + (-6.89 - 4.43i)T + (12.0 + 26.3i)T^{2} \) |
| 31 | \( 1 + (4.60 - 5.31i)T + (-4.41 - 30.6i)T^{2} \) |
| 37 | \( 1 + (-0.293 + 2.03i)T + (-35.5 - 10.4i)T^{2} \) |
| 41 | \( 1 + (1.36 + 9.46i)T + (-39.3 + 11.5i)T^{2} \) |
| 43 | \( 1 + (1.90 + 2.20i)T + (-6.11 + 42.5i)T^{2} \) |
| 47 | \( 1 - 7.43T + 47T^{2} \) |
| 53 | \( 1 + (-5.01 + 10.9i)T + (-34.7 - 40.0i)T^{2} \) |
| 59 | \( 1 + (-3.56 - 7.80i)T + (-38.6 + 44.5i)T^{2} \) |
| 61 | \( 1 + (2.15 - 2.48i)T + (-8.68 - 60.3i)T^{2} \) |
| 67 | \( 1 + (-1.61 + 0.474i)T + (56.3 - 36.2i)T^{2} \) |
| 71 | \( 1 + (-3.54 + 1.04i)T + (59.7 - 38.3i)T^{2} \) |
| 73 | \( 1 + (0.819 - 0.526i)T + (30.3 - 66.4i)T^{2} \) |
| 79 | \( 1 + (-3.11 - 6.81i)T + (-51.7 + 59.7i)T^{2} \) |
| 83 | \( 1 + (-1.05 + 7.31i)T + (-79.6 - 23.3i)T^{2} \) |
| 89 | \( 1 + (-4.11 - 4.74i)T + (-12.6 + 88.0i)T^{2} \) |
| 97 | \( 1 + (2.20 + 15.3i)T + (-93.0 + 27.3i)T^{2} \) |
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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
−10.84746425699246852948911543313, −10.23603440185692680726441707751, −9.176867298729945464202512118171, −8.693143526760611427840998596212, −7.12256800158982286407645001737, −6.75090138322270188870615151713, −5.49958205602374130868213331635, −4.21324524620794082696485806965, −3.18564246933514422642645866092, −2.12196579475667546526905241730,
0.914533218328591890090341954086, 2.26976400476022250490008861171, 3.99240068634714498262259339567, 4.64376828390830071926092701999, 6.23653361006097323214148768974, 6.72676989747959989222539925687, 8.121391493898509908740431854979, 8.823838844051142298131811626911, 9.318730132537211503752664702024, 10.55122268752669545065964476911
