| L(s) = 1 | + (−1.58 − 0.620i)2-s + (0.678 − 1.59i)3-s + (0.647 + 0.601i)4-s + (−0.761 − 0.519i)5-s + (−2.06 + 2.09i)6-s + (−0.654 − 2.56i)7-s + (0.822 + 1.70i)8-s + (−2.07 − 2.16i)9-s + (0.881 + 1.29i)10-s + (0.0278 + 0.184i)11-s + (1.39 − 0.624i)12-s + (−2.33 + 1.86i)13-s + (−0.556 + 4.45i)14-s + (−1.34 + 0.860i)15-s + (−0.372 − 4.97i)16-s + (−3.71 − 1.14i)17-s + ⋯ |
| L(s) = 1 | + (−1.11 − 0.438i)2-s + (0.391 − 0.920i)3-s + (0.323 + 0.300i)4-s + (−0.340 − 0.232i)5-s + (−0.841 + 0.856i)6-s + (−0.247 − 0.968i)7-s + (0.290 + 0.603i)8-s + (−0.692 − 0.721i)9-s + (0.278 + 0.408i)10-s + (0.00838 + 0.0556i)11-s + (0.403 − 0.180i)12-s + (−0.648 + 0.517i)13-s + (−0.148 + 1.19i)14-s + (−0.347 + 0.222i)15-s + (−0.0931 − 1.24i)16-s + (−0.901 − 0.278i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.930 + 0.366i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.930 + 0.366i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0989495 - 0.521736i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0989495 - 0.521736i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.678 + 1.59i)T \) |
| 7 | \( 1 + (0.654 + 2.56i)T \) |
| good | 2 | \( 1 + (1.58 + 0.620i)T + (1.46 + 1.36i)T^{2} \) |
| 5 | \( 1 + (0.761 + 0.519i)T + (1.82 + 4.65i)T^{2} \) |
| 11 | \( 1 + (-0.0278 - 0.184i)T + (-10.5 + 3.24i)T^{2} \) |
| 13 | \( 1 + (2.33 - 1.86i)T + (2.89 - 12.6i)T^{2} \) |
| 17 | \( 1 + (3.71 + 1.14i)T + (14.0 + 9.57i)T^{2} \) |
| 19 | \( 1 + (-2.14 + 1.24i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.363 + 1.17i)T + (-19.0 + 12.9i)T^{2} \) |
| 29 | \( 1 + (-9.18 + 2.09i)T + (26.1 - 12.5i)T^{2} \) |
| 31 | \( 1 + (-6.62 - 3.82i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-2.37 + 2.20i)T + (2.76 - 36.8i)T^{2} \) |
| 41 | \( 1 + (-5.40 + 2.60i)T + (25.5 - 32.0i)T^{2} \) |
| 43 | \( 1 + (-4.29 - 2.06i)T + (26.8 + 33.6i)T^{2} \) |
| 47 | \( 1 + (-3.79 + 9.66i)T + (-34.4 - 31.9i)T^{2} \) |
| 53 | \( 1 + (-1.45 + 1.56i)T + (-3.96 - 52.8i)T^{2} \) |
| 59 | \( 1 + (8.18 - 5.58i)T + (21.5 - 54.9i)T^{2} \) |
| 61 | \( 1 + (3.96 + 4.27i)T + (-4.55 + 60.8i)T^{2} \) |
| 67 | \( 1 + (-5.08 + 8.81i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-0.367 - 0.0838i)T + (63.9 + 30.8i)T^{2} \) |
| 73 | \( 1 + (7.84 - 3.08i)T + (53.5 - 49.6i)T^{2} \) |
| 79 | \( 1 + (-7.67 - 13.2i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-0.837 + 1.04i)T + (-18.4 - 80.9i)T^{2} \) |
| 89 | \( 1 + (5.96 + 0.899i)T + (85.0 + 26.2i)T^{2} \) |
| 97 | \( 1 + 5.60iT - 97T^{2} \) |
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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
−12.39527915551994325878099054562, −11.57245805733864215886256092946, −10.43595364280626118545510263065, −9.432907799873394665228207468795, −8.467850612846456934721809942843, −7.57058938103309933914767312660, −6.61359075994887635008186469092, −4.52428777205439779667506789779, −2.51100018582140152071462103886, −0.73227613740518694163151988777,
2.89255619687636250597859068097, 4.49568299789566185382267112793, 6.07088245989398018582746900600, 7.59800136768860437449140043060, 8.456225629575553976804054816143, 9.327760857226672254472400358603, 10.05671278476639023939477335074, 11.12995819270670404926038089980, 12.37399403547129747847164923958, 13.64846453148094617399533703580
