| L(s) = 1 | + (0.173 + 0.984i)2-s + (1.56 + 0.741i)3-s + (−0.939 + 0.342i)4-s + (−3.10 − 2.60i)5-s + (−0.458 + 1.67i)6-s + (0.144 + 0.0525i)7-s + (−0.5 − 0.866i)8-s + (1.90 + 2.32i)9-s + (2.02 − 3.50i)10-s + (0.169 − 0.141i)11-s + (−1.72 − 0.161i)12-s + (0.103 − 0.585i)13-s + (−0.0266 + 0.151i)14-s + (−2.92 − 6.37i)15-s + (0.766 − 0.642i)16-s + (−2.78 + 4.81i)17-s + ⋯ |
| L(s) = 1 | + (0.122 + 0.696i)2-s + (0.903 + 0.428i)3-s + (−0.469 + 0.171i)4-s + (−1.38 − 1.16i)5-s + (−0.187 + 0.681i)6-s + (0.0545 + 0.0198i)7-s + (−0.176 − 0.306i)8-s + (0.633 + 0.773i)9-s + (0.639 − 1.10i)10-s + (0.0510 − 0.0428i)11-s + (−0.497 − 0.0466i)12-s + (0.0286 − 0.162i)13-s + (−0.00712 + 0.0404i)14-s + (−0.754 − 1.64i)15-s + (0.191 − 0.160i)16-s + (−0.674 + 1.16i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.678 - 0.734i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.678 - 0.734i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.856819 + 0.375109i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.856819 + 0.375109i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.173 - 0.984i)T \) |
| 3 | \( 1 + (-1.56 - 0.741i)T \) |
| good | 5 | \( 1 + (3.10 + 2.60i)T + (0.868 + 4.92i)T^{2} \) |
| 7 | \( 1 + (-0.144 - 0.0525i)T + (5.36 + 4.49i)T^{2} \) |
| 11 | \( 1 + (-0.169 + 0.141i)T + (1.91 - 10.8i)T^{2} \) |
| 13 | \( 1 + (-0.103 + 0.585i)T + (-12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (2.78 - 4.81i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.91 + 3.30i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-5.50 + 2.00i)T + (17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (-0.129 - 0.736i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (4.77 - 1.73i)T + (23.7 - 19.9i)T^{2} \) |
| 37 | \( 1 + (1.87 - 3.24i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.690 + 3.91i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (-7.81 + 6.56i)T + (7.46 - 42.3i)T^{2} \) |
| 47 | \( 1 + (-0.447 - 0.162i)T + (36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 - 3.29T + 53T^{2} \) |
| 59 | \( 1 + (-5.57 - 4.67i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (3.16 + 1.15i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (-1.29 + 7.34i)T + (-62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (1.42 - 2.47i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-0.638 - 1.10i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (0.574 + 3.25i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (1.43 + 8.14i)T + (-77.9 + 28.3i)T^{2} \) |
| 89 | \( 1 + (-2.47 - 4.29i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-4.33 + 3.63i)T + (16.8 - 95.5i)T^{2} \) |
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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
−15.44735440496736777819865937373, −14.84866699221998721990625739702, −13.28979693734811477262703128645, −12.51243875365985726355157197373, −10.87443108332732787856548865640, −8.949650022568993651065485338814, −8.474823060649352226652436804198, −7.25736989628676651776713247279, −4.90213228101025682137794310419, −3.84118534107726967099610747639,
2.83059906697665274024251745673, 4.07282333162853230174924346812, 6.90254406991104027044714958256, 7.913496067823976322650479653629, 9.309317541939234348666975676203, 10.85831322591716257302553451092, 11.73676094575640623097684416171, 12.92088791134098486457444154517, 14.23110278887579713590137251060, 14.91276522726407566935888980445
