| L(s) = 1 | + (−0.104 − 0.994i)2-s + (0.918 + 1.02i)3-s + (−0.978 + 0.207i)4-s + (−1.88 + 1.20i)5-s + (0.918 − 1.02i)6-s − 0.500·7-s + (0.309 + 0.951i)8-s + (0.116 − 1.10i)9-s + (1.39 + 1.74i)10-s + (−0.917 + 0.666i)11-s + (−1.11 − 0.807i)12-s + (0.0814 − 0.774i)13-s + (0.0522 + 0.497i)14-s + (−2.96 − 0.812i)15-s + (0.913 − 0.406i)16-s + (−7.56 − 1.60i)17-s + ⋯ |
| L(s) = 1 | + (−0.0739 − 0.703i)2-s + (0.530 + 0.589i)3-s + (−0.489 + 0.103i)4-s + (−0.841 + 0.539i)5-s + (0.375 − 0.416i)6-s − 0.189·7-s + (0.109 + 0.336i)8-s + (0.0388 − 0.369i)9-s + (0.441 + 0.552i)10-s + (−0.276 + 0.201i)11-s + (−0.320 − 0.233i)12-s + (0.0225 − 0.214i)13-s + (0.0139 + 0.132i)14-s + (−0.764 − 0.209i)15-s + (0.228 − 0.101i)16-s + (−1.83 − 0.390i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.964 + 0.264i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.964 + 0.264i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0442555 - 0.328955i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0442555 - 0.328955i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.104 + 0.994i)T \) |
| 5 | \( 1 + (1.88 - 1.20i)T \) |
| 19 | \( 1 + (4.30 + 0.710i)T \) |
| good | 3 | \( 1 + (-0.918 - 1.02i)T + (-0.313 + 2.98i)T^{2} \) |
| 7 | \( 1 + 0.500T + 7T^{2} \) |
| 11 | \( 1 + (0.917 - 0.666i)T + (3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (-0.0814 + 0.774i)T + (-12.7 - 2.70i)T^{2} \) |
| 17 | \( 1 + (7.56 + 1.60i)T + (15.5 + 6.91i)T^{2} \) |
| 23 | \( 1 + (-2.44 - 1.08i)T + (15.3 + 17.0i)T^{2} \) |
| 29 | \( 1 + (-5.58 + 1.18i)T + (26.4 - 11.7i)T^{2} \) |
| 31 | \( 1 + (1.69 + 5.21i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (8.62 + 6.26i)T + (11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (3.06 - 1.36i)T + (27.4 - 30.4i)T^{2} \) |
| 43 | \( 1 + (-1.20 + 2.09i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (3.79 - 0.805i)T + (42.9 - 19.1i)T^{2} \) |
| 53 | \( 1 + (6.01 - 1.27i)T + (48.4 - 21.5i)T^{2} \) |
| 59 | \( 1 + (4.85 - 2.16i)T + (39.4 - 43.8i)T^{2} \) |
| 61 | \( 1 + (-8.12 - 3.61i)T + (40.8 + 45.3i)T^{2} \) |
| 67 | \( 1 + (3.28 - 3.64i)T + (-7.00 - 66.6i)T^{2} \) |
| 71 | \( 1 + (0.504 + 0.559i)T + (-7.42 + 70.6i)T^{2} \) |
| 73 | \( 1 + (0.459 + 4.37i)T + (-71.4 + 15.1i)T^{2} \) |
| 79 | \( 1 + (6.11 + 6.79i)T + (-8.25 + 78.5i)T^{2} \) |
| 83 | \( 1 + (-3.36 - 10.3i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (-7.22 - 3.21i)T + (59.5 + 66.1i)T^{2} \) |
| 97 | \( 1 + (3.91 + 4.34i)T + (-10.1 + 96.4i)T^{2} \) |
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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
−9.705921349658081643713709238682, −8.880754987512986259177918595965, −8.329603380978678245349976301789, −7.16244865206171078023130930752, −6.37657815886463625968825408170, −4.75551442437034863825926682556, −4.10648289541608051126882903357, −3.20473298799625790243799359320, −2.32216931988382902397548926160, −0.14279720297518715692943536920,
1.72039872811869626792278466487, 3.16470909231141123766371813318, 4.44377207445033145495682347611, 5.04946161010817041566321260211, 6.59587233650782502913239091676, 6.93406041056589440829830559305, 8.182877337957217245880234036628, 8.413529159725430599169127971026, 9.115887819755639170679297889249, 10.46283763954059398257551922092
