L(s) = 1 | + 1.49i·2-s − 2.14i·3-s − 0.247·4-s + 3.21·6-s + 0.914i·7-s + 2.62i·8-s − 1.59·9-s − 5.69·11-s + 0.529i·12-s + 5.13i·13-s − 1.37·14-s − 4.43·16-s + 5.54i·17-s − 2.38i·18-s − 4.24·19-s + ⋯ |
L(s) = 1 | + 1.05i·2-s − 1.23i·3-s − 0.123·4-s + 1.31·6-s + 0.345i·7-s + 0.929i·8-s − 0.530·9-s − 1.71·11-s + 0.152i·12-s + 1.42i·13-s − 0.366·14-s − 1.10·16-s + 1.34i·17-s − 0.562i·18-s − 0.974·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8947483287\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8947483287\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 47 | \( 1 + iT \) |
good | 2 | \( 1 - 1.49iT - 2T^{2} \) |
| 3 | \( 1 + 2.14iT - 3T^{2} \) |
| 7 | \( 1 - 0.914iT - 7T^{2} \) |
| 11 | \( 1 + 5.69T + 11T^{2} \) |
| 13 | \( 1 - 5.13iT - 13T^{2} \) |
| 17 | \( 1 - 5.54iT - 17T^{2} \) |
| 19 | \( 1 + 4.24T + 19T^{2} \) |
| 23 | \( 1 + 7.15iT - 23T^{2} \) |
| 29 | \( 1 + 1.00T + 29T^{2} \) |
| 31 | \( 1 + 6.52T + 31T^{2} \) |
| 37 | \( 1 - 10.0iT - 37T^{2} \) |
| 41 | \( 1 + 4.42T + 41T^{2} \) |
| 43 | \( 1 + 0.310iT - 43T^{2} \) |
| 53 | \( 1 - 6.38iT - 53T^{2} \) |
| 59 | \( 1 + 1.42T + 59T^{2} \) |
| 61 | \( 1 - 12.9T + 61T^{2} \) |
| 67 | \( 1 - 4.25iT - 67T^{2} \) |
| 71 | \( 1 - 3.83T + 71T^{2} \) |
| 73 | \( 1 - 9.86iT - 73T^{2} \) |
| 79 | \( 1 - 6.00T + 79T^{2} \) |
| 83 | \( 1 + 2.43iT - 83T^{2} \) |
| 89 | \( 1 + 11.8T + 89T^{2} \) |
| 97 | \( 1 + 18.0iT - 97T^{2} \) |
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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
−10.17773324576703212008311620946, −8.625015168973995132706123642264, −8.378832533187865133382671395587, −7.51863913765978532224363830319, −6.75899636674786404682069389518, −6.27828168074524744317000955215, −5.39484765504620048517742023806, −4.33618671598434252806286185694, −2.49504440988735015446668985542, −1.88295807395996835666640592141,
0.34546840690278793950712476100, 2.23344886613769224862470998448, 3.19567014770483119928143078222, 3.84852937176315755399834027882, 5.04792184022539180024785478620, 5.55146955774374873743314843180, 7.16283628321525296642764659546, 7.79429113199101101151253764660, 9.038207955480830816865371442583, 9.813218303379878618527794478647