| L(s) = 1 | + (−0.415 − 0.909i)2-s + (−1.50 + 0.442i)3-s + (−0.654 + 0.755i)4-s + (0.841 + 0.540i)5-s + (1.02 + 1.18i)6-s + (−0.440 − 3.06i)7-s + (0.959 + 0.281i)8-s + (−0.445 + 0.286i)9-s + (0.142 − 0.989i)10-s + (1.91 − 4.19i)11-s + (0.652 − 1.42i)12-s + (0.473 − 3.29i)13-s + (−2.60 + 1.67i)14-s + (−1.50 − 0.442i)15-s + (−0.142 − 0.989i)16-s + (−2.52 − 2.90i)17-s + ⋯ |
| L(s) = 1 | + (−0.293 − 0.643i)2-s + (−0.870 + 0.255i)3-s + (−0.327 + 0.377i)4-s + (0.376 + 0.241i)5-s + (0.420 + 0.484i)6-s + (−0.166 − 1.15i)7-s + (0.339 + 0.0996i)8-s + (−0.148 + 0.0954i)9-s + (0.0450 − 0.313i)10-s + (0.578 − 1.26i)11-s + (0.188 − 0.412i)12-s + (0.131 − 0.912i)13-s + (−0.695 + 0.447i)14-s + (−0.389 − 0.114i)15-s + (−0.0355 − 0.247i)16-s + (−0.611 − 0.705i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.401 + 0.915i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.401 + 0.915i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.367085 - 0.561737i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.367085 - 0.561737i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.415 + 0.909i)T \) |
| 5 | \( 1 + (-0.841 - 0.540i)T \) |
| 23 | \( 1 + (2.16 + 4.27i)T \) |
| good | 3 | \( 1 + (1.50 - 0.442i)T + (2.52 - 1.62i)T^{2} \) |
| 7 | \( 1 + (0.440 + 3.06i)T + (-6.71 + 1.97i)T^{2} \) |
| 11 | \( 1 + (-1.91 + 4.19i)T + (-7.20 - 8.31i)T^{2} \) |
| 13 | \( 1 + (-0.473 + 3.29i)T + (-12.4 - 3.66i)T^{2} \) |
| 17 | \( 1 + (2.52 + 2.90i)T + (-2.41 + 16.8i)T^{2} \) |
| 19 | \( 1 + (0.431 - 0.497i)T + (-2.70 - 18.8i)T^{2} \) |
| 29 | \( 1 + (-3.43 - 3.96i)T + (-4.12 + 28.7i)T^{2} \) |
| 31 | \( 1 + (3.49 + 1.02i)T + (26.0 + 16.7i)T^{2} \) |
| 37 | \( 1 + (1.39 - 0.895i)T + (15.3 - 33.6i)T^{2} \) |
| 41 | \( 1 + (-7.20 - 4.62i)T + (17.0 + 37.2i)T^{2} \) |
| 43 | \( 1 + (3.89 - 1.14i)T + (36.1 - 23.2i)T^{2} \) |
| 47 | \( 1 - 12.0T + 47T^{2} \) |
| 53 | \( 1 + (-0.984 - 6.84i)T + (-50.8 + 14.9i)T^{2} \) |
| 59 | \( 1 + (-1.38 + 9.64i)T + (-56.6 - 16.6i)T^{2} \) |
| 61 | \( 1 + (12.6 + 3.72i)T + (51.3 + 32.9i)T^{2} \) |
| 67 | \( 1 + (2.68 + 5.88i)T + (-43.8 + 50.6i)T^{2} \) |
| 71 | \( 1 + (-2.54 - 5.58i)T + (-46.4 + 53.6i)T^{2} \) |
| 73 | \( 1 + (6.83 - 7.88i)T + (-10.3 - 72.2i)T^{2} \) |
| 79 | \( 1 + (1.29 - 8.99i)T + (-75.7 - 22.2i)T^{2} \) |
| 83 | \( 1 + (5.93 - 3.81i)T + (34.4 - 75.4i)T^{2} \) |
| 89 | \( 1 + (-14.2 + 4.19i)T + (74.8 - 48.1i)T^{2} \) |
| 97 | \( 1 + (-5.29 - 3.40i)T + (40.2 + 88.2i)T^{2} \) |
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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
−11.58489905427874056596509433912, −10.73249482798781268376635956466, −10.48897706337753150336473461343, −9.183854688635734603914295084067, −8.065932516052939474918868044916, −6.70781753180637748448699332760, −5.67520608156060760568429708464, −4.32986312865413940893556872545, −2.98957948575559848363724664312, −0.68894187439171284872796412221,
1.92643811480943422512176732690, 4.38259843734501582882535314837, 5.65171578919422732925453132625, 6.27986363487550320557972671226, 7.26624902917650267972713279676, 8.852932949331945138463567650509, 9.273310568756344046032145133851, 10.52672465204425392673489226381, 11.84777226720236807867656437835, 12.22966098563394819569753445980
