| L(s) = 1 | + (−0.951 + 0.309i)2-s + (0.668 − 1.59i)3-s + (0.809 − 0.587i)4-s + (−0.866 + 0.5i)5-s + (−0.141 + 1.72i)6-s + (0.349 + 3.32i)7-s + (−0.587 + 0.809i)8-s + (−2.10 − 2.13i)9-s + (0.669 − 0.743i)10-s + (3.04 + 1.35i)11-s + (−0.398 − 1.68i)12-s + (−0.767 − 3.61i)13-s + (−1.35 − 3.05i)14-s + (0.220 + 1.71i)15-s + (0.309 − 0.951i)16-s + (−2.24 + 1.00i)17-s + ⋯ |
| L(s) = 1 | + (−0.672 + 0.218i)2-s + (0.385 − 0.922i)3-s + (0.404 − 0.293i)4-s + (−0.387 + 0.223i)5-s + (−0.0578 + 0.704i)6-s + (0.132 + 1.25i)7-s + (−0.207 + 0.286i)8-s + (−0.702 − 0.711i)9-s + (0.211 − 0.235i)10-s + (0.918 + 0.408i)11-s + (−0.115 − 0.486i)12-s + (−0.212 − 1.00i)13-s + (−0.363 − 0.816i)14-s + (0.0568 + 0.443i)15-s + (0.0772 − 0.237i)16-s + (−0.544 + 0.242i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.972 - 0.231i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.972 - 0.231i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.23672 + 0.145219i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.23672 + 0.145219i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.951 - 0.309i)T \) |
| 3 | \( 1 + (-0.668 + 1.59i)T \) |
| 5 | \( 1 + (0.866 - 0.5i)T \) |
| 31 | \( 1 + (1.60 - 5.33i)T \) |
| good | 7 | \( 1 + (-0.349 - 3.32i)T + (-6.84 + 1.45i)T^{2} \) |
| 11 | \( 1 + (-3.04 - 1.35i)T + (7.36 + 8.17i)T^{2} \) |
| 13 | \( 1 + (0.767 + 3.61i)T + (-11.8 + 5.28i)T^{2} \) |
| 17 | \( 1 + (2.24 - 1.00i)T + (11.3 - 12.6i)T^{2} \) |
| 19 | \( 1 + (-5.45 - 1.15i)T + (17.3 + 7.72i)T^{2} \) |
| 23 | \( 1 + (-2.39 - 1.74i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (0.463 + 1.42i)T + (-23.4 + 17.0i)T^{2} \) |
| 37 | \( 1 + (-10.2 - 5.94i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-4.72 - 4.25i)T + (4.28 + 40.7i)T^{2} \) |
| 43 | \( 1 + (2.42 - 11.4i)T + (-39.2 - 17.4i)T^{2} \) |
| 47 | \( 1 + (-7.66 - 2.49i)T + (38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (-0.935 + 8.90i)T + (-51.8 - 11.0i)T^{2} \) |
| 59 | \( 1 + (-9.40 + 8.46i)T + (6.16 - 58.6i)T^{2} \) |
| 61 | \( 1 + 10.1iT - 61T^{2} \) |
| 67 | \( 1 + (6.36 + 11.0i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-12.8 - 1.35i)T + (69.4 + 14.7i)T^{2} \) |
| 73 | \( 1 + (-1.97 + 4.42i)T + (-48.8 - 54.2i)T^{2} \) |
| 79 | \( 1 + (-0.269 - 0.604i)T + (-52.8 + 58.7i)T^{2} \) |
| 83 | \( 1 + (0.713 - 0.792i)T + (-8.67 - 82.5i)T^{2} \) |
| 89 | \( 1 + (3.28 - 2.39i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (15.6 - 11.3i)T + (29.9 - 92.2i)T^{2} \) |
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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
−9.657286398574559756934211558770, −9.260273432192692977509840625494, −8.204245846208961828516200663857, −7.82540822948089201657226741729, −6.78436720174996565939495623267, −6.10266240281009068095194490297, −5.08886558624833069345529130098, −3.34657069516634814614180421913, −2.43828831111893686382808675232, −1.17119113099630526811087944376,
0.862625578715037103666682783284, 2.55156459196281055613172081184, 4.00689453666877788386266610960, 4.15256285125694150369718481286, 5.61445399289890843076608641007, 7.07657273899694603951819082326, 7.47891412573874808101860992367, 8.740635415637863271419817972363, 9.153660345522160562182483575956, 9.919107196211414439833121903295
