| L(s) = 1 | + (0.186 − 0.982i)2-s + (0.396 + 0.00929i)3-s + (−0.930 − 0.366i)4-s + (0.0679 + 0.257i)5-s + (0.0830 − 0.387i)6-s + (−1.69 + 3.77i)7-s + (−0.533 + 0.845i)8-s + (−2.83 − 0.133i)9-s + (0.265 − 0.0187i)10-s + (−2.28 + 2.57i)11-s + (−0.365 − 0.153i)12-s + (−0.226 − 3.21i)13-s + (3.39 + 2.36i)14-s + (0.0245 + 0.102i)15-s + (0.731 + 0.681i)16-s + (−5.12 + 4.56i)17-s + ⋯ |
| L(s) = 1 | + (0.131 − 0.694i)2-s + (0.228 + 0.00536i)3-s + (−0.465 − 0.183i)4-s + (0.0303 + 0.115i)5-s + (0.0339 − 0.158i)6-s + (−0.640 + 1.42i)7-s + (−0.188 + 0.299i)8-s + (−0.946 − 0.0444i)9-s + (0.0839 − 0.00591i)10-s + (−0.689 + 0.775i)11-s + (−0.105 − 0.0444i)12-s + (−0.0627 − 0.890i)13-s + (0.906 + 0.632i)14-s + (0.00633 + 0.0265i)15-s + (0.182 + 0.170i)16-s + (−1.24 + 1.10i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.147 - 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.147 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.409137 + 0.474583i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.409137 + 0.474583i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.186 + 0.982i)T \) |
| 269 | \( 1 + (-2.66 - 16.1i)T \) |
| good | 3 | \( 1 + (-0.396 - 0.00929i)T + (2.99 + 0.140i)T^{2} \) |
| 5 | \( 1 + (-0.0679 - 0.257i)T + (-4.34 + 2.46i)T^{2} \) |
| 7 | \( 1 + (1.69 - 3.77i)T + (-4.65 - 5.23i)T^{2} \) |
| 11 | \( 1 + (2.28 - 2.57i)T + (-1.28 - 10.9i)T^{2} \) |
| 13 | \( 1 + (0.226 + 3.21i)T + (-12.8 + 1.82i)T^{2} \) |
| 17 | \( 1 + (5.12 - 4.56i)T + (1.98 - 16.8i)T^{2} \) |
| 19 | \( 1 + (1.94 - 1.28i)T + (7.37 - 17.5i)T^{2} \) |
| 23 | \( 1 + (-0.00379 + 0.161i)T + (-22.9 - 1.07i)T^{2} \) |
| 29 | \( 1 + (-1.62 + 2.86i)T + (-14.8 - 24.8i)T^{2} \) |
| 31 | \( 1 + (8.02 - 0.944i)T + (30.1 - 7.20i)T^{2} \) |
| 37 | \( 1 + (-0.464 - 0.866i)T + (-20.4 + 30.8i)T^{2} \) |
| 41 | \( 1 + (-9.81 + 1.86i)T + (38.1 - 15.0i)T^{2} \) |
| 43 | \( 1 + (-7.63 - 4.33i)T + (22.0 + 36.8i)T^{2} \) |
| 47 | \( 1 + (-11.1 + 3.22i)T + (39.7 - 25.0i)T^{2} \) |
| 53 | \( 1 + (2.54 - 3.47i)T + (-15.9 - 50.5i)T^{2} \) |
| 59 | \( 1 + (2.30 - 5.86i)T + (-43.1 - 40.2i)T^{2} \) |
| 61 | \( 1 + (8.82 - 10.9i)T + (-12.7 - 59.6i)T^{2} \) |
| 67 | \( 1 + (10.3 - 4.05i)T + (49.0 - 45.6i)T^{2} \) |
| 71 | \( 1 + (-5.22 - 7.48i)T + (-24.4 + 66.6i)T^{2} \) |
| 73 | \( 1 + (4.92 + 13.4i)T + (-55.6 + 47.2i)T^{2} \) |
| 79 | \( 1 + (-2.71 + 2.09i)T + (20.1 - 76.3i)T^{2} \) |
| 83 | \( 1 + (0.267 - 2.84i)T + (-81.5 - 15.4i)T^{2} \) |
| 89 | \( 1 + (0.976 - 5.89i)T + (-84.2 - 28.6i)T^{2} \) |
| 97 | \( 1 + (8.61 + 10.6i)T + (-20.3 + 94.8i)T^{2} \) |
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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
−10.90248398852244217714950785670, −10.40520416239987828874312255367, −9.114260421670654043408818447999, −8.799076852419416383190539828864, −7.69335126340404303379628752197, −6.12728657264621215764102434004, −5.59995942346783876512898716556, −4.26916783487268871744980120006, −2.82127081719797394304360795340, −2.32977855908999007164065603187,
0.31392843949415932648463488944, 2.75789174230684418897754616349, 3.92208760409439762723625034691, 4.93856020713840848186870559249, 6.11381854101465744424346662909, 7.01528798592504633530429397103, 7.67877846664277573347853615572, 8.959681133295956836839484542115, 9.274702327162791100510551354370, 10.84221278095966624214364585799
