| L(s) = 1 | + (0.523 − 2.15i)2-s + (0.327 − 0.945i)3-s + (−2.60 − 1.34i)4-s + (−2.17 − 0.870i)5-s + (−1.86 − 1.20i)6-s + (−1.34 − 2.27i)7-s + (−1.35 + 1.56i)8-s + (−0.786 − 0.618i)9-s + (−3.01 + 4.23i)10-s + (0.945 + 3.89i)11-s + (−2.12 + 2.02i)12-s + (0.275 + 0.603i)13-s + (−5.62 + 1.70i)14-s + (−1.53 + 1.76i)15-s + (−0.739 − 1.03i)16-s + (0.351 − 7.38i)17-s + ⋯ |
| L(s) = 1 | + (0.370 − 1.52i)2-s + (0.188 − 0.545i)3-s + (−1.30 − 0.671i)4-s + (−0.972 − 0.389i)5-s + (−0.762 − 0.490i)6-s + (−0.508 − 0.861i)7-s + (−0.478 + 0.551i)8-s + (−0.262 − 0.206i)9-s + (−0.953 + 1.33i)10-s + (0.285 + 1.17i)11-s + (−0.612 + 0.583i)12-s + (0.0764 + 0.167i)13-s + (−1.50 + 0.456i)14-s + (−0.395 + 0.457i)15-s + (−0.184 − 0.259i)16-s + (0.0853 − 1.79i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.529 - 0.848i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.529 - 0.848i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.534147 + 0.963283i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.534147 + 0.963283i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.327 + 0.945i)T \) |
| 7 | \( 1 + (1.34 + 2.27i)T \) |
| 23 | \( 1 + (-4.79 - 0.0718i)T \) |
| good | 2 | \( 1 + (-0.523 + 2.15i)T + (-1.77 - 0.916i)T^{2} \) |
| 5 | \( 1 + (2.17 + 0.870i)T + (3.61 + 3.45i)T^{2} \) |
| 11 | \( 1 + (-0.945 - 3.89i)T + (-9.77 + 5.04i)T^{2} \) |
| 13 | \( 1 + (-0.275 - 0.603i)T + (-8.51 + 9.82i)T^{2} \) |
| 17 | \( 1 + (-0.351 + 7.38i)T + (-16.9 - 1.61i)T^{2} \) |
| 19 | \( 1 + (-0.332 - 6.98i)T + (-18.9 + 1.80i)T^{2} \) |
| 29 | \( 1 + (-0.114 - 0.0733i)T + (12.0 + 26.3i)T^{2} \) |
| 31 | \( 1 + (8.14 + 1.56i)T + (28.7 + 11.5i)T^{2} \) |
| 37 | \( 1 + (-1.24 - 0.980i)T + (8.72 + 35.9i)T^{2} \) |
| 41 | \( 1 + (1.77 + 12.3i)T + (-39.3 + 11.5i)T^{2} \) |
| 43 | \( 1 + (3.56 + 4.10i)T + (-6.11 + 42.5i)T^{2} \) |
| 47 | \( 1 + (-4.50 + 7.79i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-6.01 + 0.574i)T + (52.0 - 10.0i)T^{2} \) |
| 59 | \( 1 + (-1.62 + 2.28i)T + (-19.2 - 55.7i)T^{2} \) |
| 61 | \( 1 + (-2.77 - 8.02i)T + (-47.9 + 37.7i)T^{2} \) |
| 67 | \( 1 + (5.86 + 5.59i)T + (3.18 + 66.9i)T^{2} \) |
| 71 | \( 1 + (-11.0 + 3.24i)T + (59.7 - 38.3i)T^{2} \) |
| 73 | \( 1 + (4.32 + 2.23i)T + (42.3 + 59.4i)T^{2} \) |
| 79 | \( 1 + (-16.0 - 1.53i)T + (77.5 + 14.9i)T^{2} \) |
| 83 | \( 1 + (-1.07 + 7.46i)T + (-79.6 - 23.3i)T^{2} \) |
| 89 | \( 1 + (12.4 - 2.40i)T + (82.6 - 33.0i)T^{2} \) |
| 97 | \( 1 + (1.33 + 9.27i)T + (-93.0 + 27.3i)T^{2} \) |
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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
−10.53684099173461551689340900178, −9.741076770347223950286112857010, −8.916587230065581285357490051057, −7.44251744862915370764323028816, −7.09325850006881304188897650943, −5.17756352207271955922716936198, −4.07437941149919126811046733634, −3.45497241672045321845380814161, −2.00190990595906991689978945657, −0.58752828632273175775537481938,
3.08872125171402988234805624233, 3.98314190915832520539662496585, 5.18282100181614142600059321854, 6.07433879979445500120570579230, 6.85175620081486798849198486777, 7.976193492321352920525155018045, 8.611659501914689386091923850532, 9.312965168180676928827270598999, 10.90772533638593468233515396057, 11.34260622014539234253881394242
