| L(s) = 1 | + (−0.373 − 0.818i)2-s + (−0.841 − 0.540i)3-s + (0.779 − 0.899i)4-s + (−0.297 − 2.07i)5-s + (−0.128 + 0.890i)6-s + (0.329 + 0.721i)7-s + (−2.75 − 0.808i)8-s + (0.415 + 0.909i)9-s + (−1.58 + 1.01i)10-s + (−0.173 − 1.20i)11-s + (−1.14 + 0.335i)12-s + (−1.11 + 0.327i)13-s + (0.467 − 0.539i)14-s + (−0.869 + 1.90i)15-s + (0.0285 + 0.198i)16-s + (−1.51 − 1.74i)17-s + ⋯ |
| L(s) = 1 | + (−0.264 − 0.578i)2-s + (−0.485 − 0.312i)3-s + (0.389 − 0.449i)4-s + (−0.133 − 0.926i)5-s + (−0.0522 + 0.363i)6-s + (0.124 + 0.272i)7-s + (−0.973 − 0.285i)8-s + (0.138 + 0.303i)9-s + (−0.500 + 0.321i)10-s + (−0.0521 − 0.362i)11-s + (−0.329 + 0.0968i)12-s + (−0.309 + 0.0907i)13-s + (0.124 − 0.144i)14-s + (−0.224 + 0.491i)15-s + (0.00713 + 0.0496i)16-s + (−0.366 − 0.423i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.779 + 0.625i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.779 + 0.625i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.289926 - 0.824317i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.289926 - 0.824317i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.841 + 0.540i)T \) |
| 67 | \( 1 + (7.72 - 2.70i)T \) |
| good | 2 | \( 1 + (0.373 + 0.818i)T + (-1.30 + 1.51i)T^{2} \) |
| 5 | \( 1 + (0.297 + 2.07i)T + (-4.79 + 1.40i)T^{2} \) |
| 7 | \( 1 + (-0.329 - 0.721i)T + (-4.58 + 5.29i)T^{2} \) |
| 11 | \( 1 + (0.173 + 1.20i)T + (-10.5 + 3.09i)T^{2} \) |
| 13 | \( 1 + (1.11 - 0.327i)T + (10.9 - 7.02i)T^{2} \) |
| 17 | \( 1 + (1.51 + 1.74i)T + (-2.41 + 16.8i)T^{2} \) |
| 19 | \( 1 + (-0.224 + 0.490i)T + (-12.4 - 14.3i)T^{2} \) |
| 23 | \( 1 + (2.65 + 1.70i)T + (9.55 + 20.9i)T^{2} \) |
| 29 | \( 1 - 8.92T + 29T^{2} \) |
| 31 | \( 1 + (-0.336 - 0.0987i)T + (26.0 + 16.7i)T^{2} \) |
| 37 | \( 1 - 5.06T + 37T^{2} \) |
| 41 | \( 1 + (-0.0971 - 0.112i)T + (-5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (-6.36 - 7.34i)T + (-6.11 + 42.5i)T^{2} \) |
| 47 | \( 1 + (-2.14 - 1.38i)T + (19.5 + 42.7i)T^{2} \) |
| 53 | \( 1 + (-3.13 + 3.61i)T + (-7.54 - 52.4i)T^{2} \) |
| 59 | \( 1 + (-6.85 - 2.01i)T + (49.6 + 31.8i)T^{2} \) |
| 61 | \( 1 + (-0.586 + 4.07i)T + (-58.5 - 17.1i)T^{2} \) |
| 71 | \( 1 + (0.880 - 1.01i)T + (-10.1 - 70.2i)T^{2} \) |
| 73 | \( 1 + (-2.12 + 14.7i)T + (-70.0 - 20.5i)T^{2} \) |
| 79 | \( 1 + (7.51 - 2.20i)T + (66.4 - 42.7i)T^{2} \) |
| 83 | \( 1 + (-1.82 - 12.6i)T + (-79.6 + 23.3i)T^{2} \) |
| 89 | \( 1 + (3.77 - 2.42i)T + (36.9 - 80.9i)T^{2} \) |
| 97 | \( 1 + 16.1T + 97T^{2} \) |
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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
−12.00963043083777663288585671433, −11.22409728460554253066047064896, −10.23673215375272330441259021015, −9.210748368349998383783217368594, −8.228859058258566085295471959534, −6.77745937389061761960526596172, −5.73309318002568255305811042441, −4.60859043393791883046476123456, −2.53956635842980170969260414064, −0.896377656176402001976512501516,
2.72295219408381214069161575005, 4.16701114516852135830641106579, 5.81398901117686449627081504933, 6.81330294406741292609579169040, 7.53111528682353466505161990329, 8.693622221071661854907518946780, 10.05105202154630862841352558501, 10.86018683187705599247869285398, 11.77855806389468724923788196682, 12.58936433981107629287222507757
