| L(s) = 1 | + (0.173 − 0.984i)2-s + (0.140 + 1.72i)3-s + (−0.939 − 0.342i)4-s + (2.42 − 2.03i)5-s + (1.72 + 0.161i)6-s + (−3.46 + 1.26i)7-s + (−0.5 + 0.866i)8-s + (−2.96 + 0.484i)9-s + (−1.58 − 2.74i)10-s + (−1.75 − 1.46i)11-s + (0.458 − 1.67i)12-s + (0.538 + 3.05i)13-s + (0.640 + 3.62i)14-s + (3.85 + 3.90i)15-s + (0.766 + 0.642i)16-s + (−0.862 − 1.49i)17-s + ⋯ |
| L(s) = 1 | + (0.122 − 0.696i)2-s + (0.0810 + 0.996i)3-s + (−0.469 − 0.171i)4-s + (1.08 − 0.910i)5-s + (0.704 + 0.0659i)6-s + (−1.30 + 0.476i)7-s + (−0.176 + 0.306i)8-s + (−0.986 + 0.161i)9-s + (−0.500 − 0.867i)10-s + (−0.527 − 0.442i)11-s + (0.132 − 0.482i)12-s + (0.149 + 0.846i)13-s + (0.171 + 0.970i)14-s + (0.995 + 1.00i)15-s + (0.191 + 0.160i)16-s + (−0.209 − 0.362i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.943 + 0.331i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.943 + 0.331i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.879971 - 0.150256i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.879971 - 0.150256i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.140 - 1.72i)T \) |
| good | 5 | \( 1 + (-2.42 + 2.03i)T + (0.868 - 4.92i)T^{2} \) |
| 7 | \( 1 + (3.46 - 1.26i)T + (5.36 - 4.49i)T^{2} \) |
| 11 | \( 1 + (1.75 + 1.46i)T + (1.91 + 10.8i)T^{2} \) |
| 13 | \( 1 + (-0.538 - 3.05i)T + (-12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (0.862 + 1.49i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.69 + 2.93i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.15 - 1.14i)T + (17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (0.101 - 0.576i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (-4.35 - 1.58i)T + (23.7 + 19.9i)T^{2} \) |
| 37 | \( 1 + (-3.65 - 6.32i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (1.22 + 6.97i)T + (-38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (-1.27 - 1.06i)T + (7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (-3.61 + 1.31i)T + (36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 + 2.58T + 53T^{2} \) |
| 59 | \( 1 + (7.40 - 6.21i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (12.3 - 4.47i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (1.49 + 8.47i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (0.993 + 1.72i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (5.32 - 9.22i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-2.44 + 13.8i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (0.538 - 3.05i)T + (-77.9 - 28.3i)T^{2} \) |
| 89 | \( 1 + (-8.67 + 15.0i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-7.04 - 5.90i)T + (16.8 + 95.5i)T^{2} \) |
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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
−15.49676549349205841562086732077, −13.82015952008156761400075874094, −13.21040081723104218635661320721, −11.89012543446478273232208449329, −10.45998674048843389824743834186, −9.356336908891079925551746874955, −8.981264507000523657803178351288, −6.03483894661563278655031822731, −4.83378555322537888982443692768, −2.91630507369118712360623845274,
2.95074388022841310194526929879, 5.86887439080793451778033588558, 6.63577943667956578778559378443, 7.76091571887258369058251547947, 9.530707812889934449300478142768, 10.58785746035975508297960593748, 12.63598743506795851661650962670, 13.29752534848387743154550918860, 14.15353192393420481459468555389, 15.28588607784934365085366782655
