| L(s) = 1 | + (0.978 + 0.207i)2-s + (−0.104 + 0.994i)3-s + (0.913 + 0.406i)4-s + (2.40 + 2.66i)5-s + (−0.309 + 0.951i)6-s + (1.36 − 2.26i)7-s + (0.809 + 0.587i)8-s + (−0.978 − 0.207i)9-s + (1.79 + 3.10i)10-s + (3.31 + 0.119i)11-s + (−0.5 + 0.866i)12-s + (−1.81 − 5.60i)13-s + (1.80 − 1.93i)14-s + (−2.90 + 2.11i)15-s + (0.669 + 0.743i)16-s + (−6.24 + 1.32i)17-s + ⋯ |
| L(s) = 1 | + (0.691 + 0.147i)2-s + (−0.0603 + 0.574i)3-s + (0.456 + 0.203i)4-s + (1.07 + 1.19i)5-s + (−0.126 + 0.388i)6-s + (0.517 − 0.855i)7-s + (0.286 + 0.207i)8-s + (−0.326 − 0.0693i)9-s + (0.567 + 0.983i)10-s + (0.999 + 0.0361i)11-s + (−0.144 + 0.250i)12-s + (−0.504 − 1.55i)13-s + (0.483 − 0.516i)14-s + (−0.750 + 0.544i)15-s + (0.167 + 0.185i)16-s + (−1.51 + 0.322i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.532 - 0.846i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.532 - 0.846i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.16907 + 1.19755i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.16907 + 1.19755i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.978 - 0.207i)T \) |
| 3 | \( 1 + (0.104 - 0.994i)T \) |
| 7 | \( 1 + (-1.36 + 2.26i)T \) |
| 11 | \( 1 + (-3.31 - 0.119i)T \) |
| good | 5 | \( 1 + (-2.40 - 2.66i)T + (-0.522 + 4.97i)T^{2} \) |
| 13 | \( 1 + (1.81 + 5.60i)T + (-10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (6.24 - 1.32i)T + (15.5 - 6.91i)T^{2} \) |
| 19 | \( 1 + (1.88 - 0.839i)T + (12.7 - 14.1i)T^{2} \) |
| 23 | \( 1 + (3.55 - 6.14i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-7.40 + 5.38i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (0.752 - 0.835i)T + (-3.24 - 30.8i)T^{2} \) |
| 37 | \( 1 + (0.370 + 3.52i)T + (-36.1 + 7.69i)T^{2} \) |
| 41 | \( 1 + (6.19 + 4.49i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + 4.49T + 43T^{2} \) |
| 47 | \( 1 + (1.09 - 0.487i)T + (31.4 - 34.9i)T^{2} \) |
| 53 | \( 1 + (0.286 - 0.317i)T + (-5.54 - 52.7i)T^{2} \) |
| 59 | \( 1 + (-0.941 - 0.419i)T + (39.4 + 43.8i)T^{2} \) |
| 61 | \( 1 + (-6.11 - 6.79i)T + (-6.37 + 60.6i)T^{2} \) |
| 67 | \( 1 + (-1.85 - 3.20i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-1.36 + 4.21i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (8.15 + 3.63i)T + (48.8 + 54.2i)T^{2} \) |
| 79 | \( 1 + (0.378 + 0.0804i)T + (72.1 + 32.1i)T^{2} \) |
| 83 | \( 1 + (3.50 - 10.7i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (-6.39 + 11.0i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-3.48 - 10.7i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−11.03316730183970785156376862860, −10.36932129008123727519728172217, −9.820895621066590643483130988219, −8.408153053198884235984147552662, −7.22141410534580614536256482723, −6.41826504920423494557488038491, −5.57215984844724413013240701905, −4.37276561242360572330249400608, −3.35708263954851005144619149928, −2.08463545729109004926845704054,
1.62923229188451396636794453108, 2.32164409635974876510343756686, 4.50706864981496621336072210678, 4.96144461113655750694553010513, 6.35243309447365730284979658753, 6.65349629867766583629634235857, 8.585410830360219547818376620872, 8.876445497131982379788147389810, 9.914321607473699927644721220216, 11.34227024000020291501591897181
