L(s) = 1 | + (0.780 + 2.40i)2-s + (1.77 + 1.28i)3-s + (−3.54 + 2.57i)4-s + (0.678 − 2.08i)5-s + (−1.71 + 5.26i)6-s + (0.932 − 0.677i)7-s + (−4.85 − 3.52i)8-s + (0.556 + 1.71i)9-s + 5.54·10-s + (−3.01 − 1.38i)11-s − 9.59·12-s + (−0.288 − 0.887i)13-s + (2.35 + 1.71i)14-s + (3.89 − 2.82i)15-s + (1.97 − 6.09i)16-s + (0.309 − 0.951i)17-s + ⋯ |
L(s) = 1 | + (0.551 + 1.69i)2-s + (1.02 + 0.743i)3-s + (−1.77 + 1.28i)4-s + (0.303 − 0.933i)5-s + (−0.698 + 2.14i)6-s + (0.352 − 0.256i)7-s + (−1.71 − 1.24i)8-s + (0.185 + 0.571i)9-s + 1.75·10-s + (−0.908 − 0.417i)11-s − 2.76·12-s + (−0.0799 − 0.246i)13-s + (0.629 + 0.457i)14-s + (1.00 − 0.729i)15-s + (0.494 − 1.52i)16-s + (0.0749 − 0.230i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.703 - 0.710i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.703 - 0.710i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.722513 + 1.73207i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.722513 + 1.73207i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 + (3.01 + 1.38i)T \) |
| 17 | \( 1 + (-0.309 + 0.951i)T \) |
good | 2 | \( 1 + (-0.780 - 2.40i)T + (-1.61 + 1.17i)T^{2} \) |
| 3 | \( 1 + (-1.77 - 1.28i)T + (0.927 + 2.85i)T^{2} \) |
| 5 | \( 1 + (-0.678 + 2.08i)T + (-4.04 - 2.93i)T^{2} \) |
| 7 | \( 1 + (-0.932 + 0.677i)T + (2.16 - 6.65i)T^{2} \) |
| 13 | \( 1 + (0.288 + 0.887i)T + (-10.5 + 7.64i)T^{2} \) |
| 19 | \( 1 + (4.23 + 3.08i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 - 5.52T + 23T^{2} \) |
| 29 | \( 1 + (6.60 - 4.80i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (-2.19 - 6.76i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (8.81 - 6.40i)T + (11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (-6.89 - 5.00i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + 2.67T + 43T^{2} \) |
| 47 | \( 1 + (-2.54 - 1.84i)T + (14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (3.15 + 9.71i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (-5.17 + 3.75i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (0.628 - 1.93i)T + (-49.3 - 35.8i)T^{2} \) |
| 67 | \( 1 - 12.3T + 67T^{2} \) |
| 71 | \( 1 + (-4.48 + 13.7i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (-0.700 + 0.508i)T + (22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (1.06 + 3.28i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (-1.31 + 4.03i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + 6.94T + 89T^{2} \) |
| 97 | \( 1 + (-0.177 - 0.547i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−13.27108975489243047001364714001, −12.72629342417538786066814328009, −10.77059989672180157164682579968, −9.337104562494689690570737335611, −8.666480150897242378953022362351, −8.019930842578393130410052833869, −6.77660671952726149816256882302, −5.22371197667182485952687258371, −4.72985090407223239288870039379, −3.29300477794381158134689266422,
2.04937498765869809712094434428, 2.58394281781140736914346656772, 3.92066606265822806233176828376, 5.50228772333690363769786190344, 7.17922705233516523575415866427, 8.386449633795990853528005545772, 9.515400397835772168250716847154, 10.53603307061441206683541817810, 11.18334233495850166163580119742, 12.46716018740517729289858842417