L(s) = 1 | + (−1 + i)3-s + (0.386 − 2.20i)5-s + (−0.386 + 0.386i)7-s + i·9-s − 0.772i·11-s + (2.70 − 2.70i)13-s + (1.81 + 2.58i)15-s + (4.79 − 4.79i)17-s + 5.95·19-s − 0.772i·21-s + (−2.82 + 3.87i)23-s + (−4.70 − 1.70i)25-s + (−4 − 4i)27-s + 2.29i·29-s + 9.10·31-s + ⋯ |
L(s) = 1 | + (−0.577 + 0.577i)3-s + (0.172 − 0.984i)5-s + (−0.146 + 0.146i)7-s + 0.333i·9-s − 0.232i·11-s + (0.749 − 0.749i)13-s + (0.468 + 0.668i)15-s + (1.16 − 1.16i)17-s + 1.36·19-s − 0.168i·21-s + (−0.589 + 0.807i)23-s + (−0.940 − 0.340i)25-s + (−0.769 − 0.769i)27-s + 0.426i·29-s + 1.63·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 + 0.248i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.968 + 0.248i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.22399 - 0.154402i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.22399 - 0.154402i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.386 + 2.20i)T \) |
| 23 | \( 1 + (2.82 - 3.87i)T \) |
good | 3 | \( 1 + (1 - i)T - 3iT^{2} \) |
| 7 | \( 1 + (0.386 - 0.386i)T - 7iT^{2} \) |
| 11 | \( 1 + 0.772iT - 11T^{2} \) |
| 13 | \( 1 + (-2.70 + 2.70i)T - 13iT^{2} \) |
| 17 | \( 1 + (-4.79 + 4.79i)T - 17iT^{2} \) |
| 19 | \( 1 - 5.95T + 19T^{2} \) |
| 29 | \( 1 - 2.29iT - 29T^{2} \) |
| 31 | \( 1 - 9.10T + 31T^{2} \) |
| 37 | \( 1 + (-4.79 + 4.79i)T - 37iT^{2} \) |
| 41 | \( 1 - 7.70T + 41T^{2} \) |
| 43 | \( 1 + (-4.13 - 4.13i)T + 43iT^{2} \) |
| 47 | \( 1 + (6.40 + 6.40i)T + 47iT^{2} \) |
| 53 | \( 1 + (5.56 + 5.56i)T + 53iT^{2} \) |
| 59 | \( 1 - 3.70iT - 59T^{2} \) |
| 61 | \( 1 + 7.49iT - 61T^{2} \) |
| 67 | \( 1 + (6.33 - 6.33i)T - 67iT^{2} \) |
| 71 | \( 1 - 2.29T + 71T^{2} \) |
| 73 | \( 1 + (2.40 - 2.40i)T - 73iT^{2} \) |
| 79 | \( 1 + 13.2T + 79T^{2} \) |
| 83 | \( 1 + (-0.386 - 0.386i)T + 83iT^{2} \) |
| 89 | \( 1 + 16.8T + 89T^{2} \) |
| 97 | \( 1 + (10.6 - 10.6i)T - 97iT^{2} \) |
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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
−11.13725506277153070060824831253, −9.884598382628762273576008865902, −9.571272767618589721001558123275, −8.266463106149668130906300040718, −7.56573596224231138901024505956, −5.83059149133602630160807481472, −5.44021415065189335392873701970, −4.42231288597151090460020337987, −3.05525358147149110767706438538, −1.02291089505521127061771939610,
1.34343477883820432084415182863, 3.01756611848481441741763060894, 4.16314779677423452288750425460, 5.90857760941800479134504661942, 6.29206176517167591028956230310, 7.26609132471157080658909460835, 8.156634827773783102277618184594, 9.556704596301339265427883262678, 10.21131439187031566606629642566, 11.23798393126067094439424077202