| 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
−13.21068297280112409283705643228, −12.01311710611681777396366007652, −11.09384233647394317334693756637, −10.39040126934015554778731568826, −8.937186140849734773939590296123, −8.060834288778244218455023051501, −6.96740797888109357442775756365, −5.97325714320644607015876095958, −3.42794836077806491681746206815, −2.29985399925755402952844568343,
1.35833613407152544146735100438, 4.10396741561284494546443903544, 5.00404828242109854314901002784, 6.89583889860669784812160253438, 8.088035427136348709071970178188, 9.009410870917194679590409921454, 9.565870351374508095338738998140, 11.07827893178160267872070671357, 11.84374990113629163121984298733, 13.32956525782578287215903225014
