L(s) = 1 | + (−0.932 − 0.361i)4-s + (−0.156 + 0.0971i)7-s + (1.58 − 0.981i)13-s + (0.739 + 0.673i)16-s + (−0.111 − 1.20i)19-s + (−0.273 − 0.961i)25-s + (0.181 − 0.0339i)28-s + (1.20 − 1.32i)31-s + (0.942 + 0.469i)37-s + (−1.72 + 0.857i)43-s + (−0.430 + 0.864i)49-s + (−1.83 + 0.342i)52-s + (0.404 + 1.42i)61-s + (−0.445 − 0.895i)64-s + (0.193 − 0.312i)67-s + ⋯ |
L(s) = 1 | + (−0.932 − 0.361i)4-s + (−0.156 + 0.0971i)7-s + (1.58 − 0.981i)13-s + (0.739 + 0.673i)16-s + (−0.111 − 1.20i)19-s + (−0.273 − 0.961i)25-s + (0.181 − 0.0339i)28-s + (1.20 − 1.32i)31-s + (0.942 + 0.469i)37-s + (−1.72 + 0.857i)43-s + (−0.430 + 0.864i)49-s + (−1.83 + 0.342i)52-s + (0.404 + 1.42i)61-s + (−0.445 − 0.895i)64-s + (0.193 − 0.312i)67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.705 + 0.708i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.705 + 0.708i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8298119426\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8298119426\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 103 | \( 1 + (0.445 - 0.895i)T \) |
good | 2 | \( 1 + (0.932 + 0.361i)T^{2} \) |
| 5 | \( 1 + (0.273 + 0.961i)T^{2} \) |
| 7 | \( 1 + (0.156 - 0.0971i)T + (0.445 - 0.895i)T^{2} \) |
| 11 | \( 1 + (-0.932 - 0.361i)T^{2} \) |
| 13 | \( 1 + (-1.58 + 0.981i)T + (0.445 - 0.895i)T^{2} \) |
| 17 | \( 1 + (-0.850 + 0.526i)T^{2} \) |
| 19 | \( 1 + (0.111 + 1.20i)T + (-0.982 + 0.183i)T^{2} \) |
| 23 | \( 1 + (0.932 - 0.361i)T^{2} \) |
| 29 | \( 1 + (-0.273 - 0.961i)T^{2} \) |
| 31 | \( 1 + (-1.20 + 1.32i)T + (-0.0922 - 0.995i)T^{2} \) |
| 37 | \( 1 + (-0.942 - 0.469i)T + (0.602 + 0.798i)T^{2} \) |
| 41 | \( 1 + (-0.273 + 0.961i)T^{2} \) |
| 43 | \( 1 + (1.72 - 0.857i)T + (0.602 - 0.798i)T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + (0.982 - 0.183i)T^{2} \) |
| 59 | \( 1 + (0.445 + 0.895i)T^{2} \) |
| 61 | \( 1 + (-0.404 - 1.42i)T + (-0.850 + 0.526i)T^{2} \) |
| 67 | \( 1 + (-0.193 + 0.312i)T + (-0.445 - 0.895i)T^{2} \) |
| 71 | \( 1 + (0.273 - 0.961i)T^{2} \) |
| 73 | \( 1 + (-1.07 + 0.811i)T + (0.273 - 0.961i)T^{2} \) |
| 79 | \( 1 + (1.02 - 1.35i)T + (-0.273 - 0.961i)T^{2} \) |
| 83 | \( 1 + (0.445 - 0.895i)T^{2} \) |
| 89 | \( 1 + (-0.739 + 0.673i)T^{2} \) |
| 97 | \( 1 + (0.149 - 0.526i)T + (-0.850 - 0.526i)T^{2} \) |
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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
−10.06899291454981678795289174480, −9.389625333909602306566853740904, −8.416081590395662980686403116191, −8.024965789189925357515082914831, −6.48628228586436703678799008587, −5.89438121522051994619245398503, −4.83519450883020716948212381789, −3.97844203097580855679799707512, −2.82331257116847354831222449366, −0.980549395490329834323933257357,
1.51769311471812165477079952874, 3.38049933342539756970711206691, 3.97295306323785208629617613051, 5.04767693999324058500847702869, 6.06765146263616835040649015434, 6.96309192607546030467932270181, 8.181526287635128282842626180718, 8.592645661437697591709884519550, 9.505311080923600599387497269632, 10.21925301312595434758261576933