| L(s) = 1 | + (0.911 + 0.411i)2-s + (−1.06 + 2.27i)3-s + (0.660 + 0.750i)4-s + (−2.01 + 0.463i)5-s + (−1.91 + 1.63i)6-s + (1.89 + 3.07i)7-s + (0.292 + 0.956i)8-s + (−2.12 − 2.55i)9-s + (−2.02 − 0.406i)10-s + (−0.268 − 1.44i)11-s + (−2.41 + 0.702i)12-s + (0.217 + 0.0184i)13-s + (0.458 + 3.57i)14-s + (1.09 − 5.07i)15-s + (−0.127 + 0.991i)16-s + (−6.62 − 2.34i)17-s + ⋯ |
| L(s) = 1 | + (0.644 + 0.291i)2-s + (−0.616 + 1.31i)3-s + (0.330 + 0.375i)4-s + (−0.899 + 0.207i)5-s + (−0.779 + 0.667i)6-s + (0.715 + 1.16i)7-s + (0.103 + 0.338i)8-s + (−0.707 − 0.851i)9-s + (−0.639 − 0.128i)10-s + (−0.0808 − 0.434i)11-s + (−0.696 + 0.202i)12-s + (0.0602 + 0.00512i)13-s + (0.122 + 0.956i)14-s + (0.282 − 1.30i)15-s + (−0.0317 + 0.247i)16-s + (−1.60 − 0.568i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 446 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 - 0.138i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 446 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.990 - 0.138i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0885049 + 1.27441i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0885049 + 1.27441i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.911 - 0.411i)T \) |
| 223 | \( 1 + (-5.96 + 13.6i)T \) |
| good | 3 | \( 1 + (1.06 - 2.27i)T + (-1.91 - 2.30i)T^{2} \) |
| 5 | \( 1 + (2.01 - 0.463i)T + (4.49 - 2.18i)T^{2} \) |
| 7 | \( 1 + (-1.89 - 3.07i)T + (-3.15 + 6.25i)T^{2} \) |
| 11 | \( 1 + (0.268 + 1.44i)T + (-10.2 + 3.95i)T^{2} \) |
| 13 | \( 1 + (-0.217 - 0.0184i)T + (12.8 + 2.19i)T^{2} \) |
| 17 | \( 1 + (6.62 + 2.34i)T + (13.2 + 10.6i)T^{2} \) |
| 19 | \( 1 + (-3.87 - 0.219i)T + (18.8 + 2.14i)T^{2} \) |
| 23 | \( 1 + (-0.165 - 0.177i)T + (-1.62 + 22.9i)T^{2} \) |
| 29 | \( 1 + (-3.15 + 1.32i)T + (20.3 - 20.6i)T^{2} \) |
| 31 | \( 1 + (-0.419 - 2.67i)T + (-29.5 + 9.49i)T^{2} \) |
| 37 | \( 1 + (-5.75 - 4.91i)T + (5.73 + 36.5i)T^{2} \) |
| 41 | \( 1 + (-0.280 - 6.60i)T + (-40.8 + 3.47i)T^{2} \) |
| 43 | \( 1 + (0.0876 + 1.23i)T + (-42.5 + 6.06i)T^{2} \) |
| 47 | \( 1 + (2.84 - 0.161i)T + (46.6 - 5.30i)T^{2} \) |
| 53 | \( 1 + (-8.26 - 4.93i)T + (25.1 + 46.6i)T^{2} \) |
| 59 | \( 1 + (-2.82 + 2.70i)T + (2.50 - 58.9i)T^{2} \) |
| 61 | \( 1 + (-2.80 + 4.02i)T + (-21.1 - 57.2i)T^{2} \) |
| 67 | \( 1 + (-12.5 + 2.52i)T + (61.8 - 25.8i)T^{2} \) |
| 71 | \( 1 + (0.990 + 4.03i)T + (-62.9 + 32.8i)T^{2} \) |
| 73 | \( 1 + (4.64 - 10.6i)T + (-49.7 - 53.4i)T^{2} \) |
| 79 | \( 1 + (8.84 - 5.62i)T + (33.5 - 71.5i)T^{2} \) |
| 83 | \( 1 + (1.07 - 15.2i)T + (-82.1 - 11.7i)T^{2} \) |
| 89 | \( 1 + (-1.85 - 0.427i)T + (80.0 + 38.9i)T^{2} \) |
| 97 | \( 1 + (5.62 - 6.03i)T + (-6.85 - 96.7i)T^{2} \) |
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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
−11.41528800457207477784571487725, −11.10928006593371955309406065658, −9.799416281070182859721401766298, −8.789132008941968078508241058392, −7.960616523667916450782086131535, −6.64128771383703698649382588344, −5.51110434861699416335371361748, −4.83580099357816391724882257097, −3.99191104377655662124973718873, −2.74651876688075913284408078504,
0.71256545315674180103957048314, 2.04703694305387869085299939476, 3.93346315244173030410212237058, 4.70436954740305527373706025092, 6.01131849215317941444630237672, 7.11997262214851275979623982577, 7.47887016733391270834896779086, 8.541707689789784084379438249721, 10.17400021910556904898651024161, 11.22901031611894963689173783859
