| L(s) = 1 | + (−0.959 + 0.281i)2-s + (0.654 + 0.755i)3-s + (0.841 − 0.540i)4-s + (−0.544 − 3.78i)5-s + (−0.841 − 0.540i)6-s + (1.20 − 2.63i)7-s + (−0.654 + 0.755i)8-s + (−0.142 + 0.989i)9-s + (1.58 + 3.47i)10-s + (1.51 + 0.445i)11-s + (0.959 + 0.281i)12-s + (1.95 + 4.28i)13-s + (−0.411 + 2.86i)14-s + (2.50 − 2.88i)15-s + (0.415 − 0.909i)16-s + (−3.98 − 2.56i)17-s + ⋯ |
| L(s) = 1 | + (−0.678 + 0.199i)2-s + (0.378 + 0.436i)3-s + (0.420 − 0.270i)4-s + (−0.243 − 1.69i)5-s + (−0.343 − 0.220i)6-s + (0.454 − 0.994i)7-s + (−0.231 + 0.267i)8-s + (−0.0474 + 0.329i)9-s + (0.502 + 1.09i)10-s + (0.457 + 0.134i)11-s + (0.276 + 0.0813i)12-s + (0.542 + 1.18i)13-s + (−0.110 + 0.765i)14-s + (0.646 − 0.746i)15-s + (0.103 − 0.227i)16-s + (−0.966 − 0.621i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.868 + 0.495i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.868 + 0.495i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.887138 - 0.235495i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.887138 - 0.235495i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.959 - 0.281i)T \) |
| 3 | \( 1 + (-0.654 - 0.755i)T \) |
| 23 | \( 1 + (-2.88 - 3.82i)T \) |
| good | 5 | \( 1 + (0.544 + 3.78i)T + (-4.79 + 1.40i)T^{2} \) |
| 7 | \( 1 + (-1.20 + 2.63i)T + (-4.58 - 5.29i)T^{2} \) |
| 11 | \( 1 + (-1.51 - 0.445i)T + (9.25 + 5.94i)T^{2} \) |
| 13 | \( 1 + (-1.95 - 4.28i)T + (-8.51 + 9.82i)T^{2} \) |
| 17 | \( 1 + (3.98 + 2.56i)T + (7.06 + 15.4i)T^{2} \) |
| 19 | \( 1 + (-1.80 + 1.16i)T + (7.89 - 17.2i)T^{2} \) |
| 29 | \( 1 + (0.495 + 0.318i)T + (12.0 + 26.3i)T^{2} \) |
| 31 | \( 1 + (0.111 - 0.129i)T + (-4.41 - 30.6i)T^{2} \) |
| 37 | \( 1 + (0.685 - 4.76i)T + (-35.5 - 10.4i)T^{2} \) |
| 41 | \( 1 + (-1.47 - 10.2i)T + (-39.3 + 11.5i)T^{2} \) |
| 43 | \( 1 + (8.03 + 9.27i)T + (-6.11 + 42.5i)T^{2} \) |
| 47 | \( 1 - 5.45T + 47T^{2} \) |
| 53 | \( 1 + (4.28 - 9.37i)T + (-34.7 - 40.0i)T^{2} \) |
| 59 | \( 1 + (0.790 + 1.73i)T + (-38.6 + 44.5i)T^{2} \) |
| 61 | \( 1 + (-8.28 + 9.56i)T + (-8.68 - 60.3i)T^{2} \) |
| 67 | \( 1 + (5.63 - 1.65i)T + (56.3 - 36.2i)T^{2} \) |
| 71 | \( 1 + (-11.3 + 3.33i)T + (59.7 - 38.3i)T^{2} \) |
| 73 | \( 1 + (11.1 - 7.14i)T + (30.3 - 66.4i)T^{2} \) |
| 79 | \( 1 + (-4.85 - 10.6i)T + (-51.7 + 59.7i)T^{2} \) |
| 83 | \( 1 + (-0.0161 + 0.112i)T + (-79.6 - 23.3i)T^{2} \) |
| 89 | \( 1 + (-8.01 - 9.24i)T + (-12.6 + 88.0i)T^{2} \) |
| 97 | \( 1 + (0.124 + 0.864i)T + (-93.0 + 27.3i)T^{2} \) |
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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
−13.32956525782578287215903225014, −11.84374990113629163121984298733, −11.07827893178160267872070671357, −9.565870351374508095338738998140, −9.009410870917194679590409921454, −8.088035427136348709071970178188, −6.89583889860669784812160253438, −5.00404828242109854314901002784, −4.10396741561284494546443903544, −1.35833613407152544146735100438,
2.29985399925755402952844568343, 3.42794836077806491681746206815, 5.97325714320644607015876095958, 6.96740797888109357442775756365, 8.060834288778244218455023051501, 8.937186140849734773939590296123, 10.39040126934015554778731568826, 11.09384233647394317334693756637, 12.01311710611681777396366007652, 13.21068297280112409283705643228
