| L(s) = 1 | − 3-s + 4·7-s + 9-s + 4·11-s + 4·13-s − 4·21-s − 4·23-s − 27-s + 6·29-s + 4·31-s − 4·33-s − 2·37-s − 4·39-s + 2·41-s − 43-s − 8·47-s + 9·49-s − 6·53-s − 4·59-s + 4·61-s + 4·63-s − 8·67-s + 4·69-s − 4·71-s + 10·73-s + 16·77-s + 8·79-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.51·7-s + 1/3·9-s + 1.20·11-s + 1.10·13-s − 0.872·21-s − 0.834·23-s − 0.192·27-s + 1.11·29-s + 0.718·31-s − 0.696·33-s − 0.328·37-s − 0.640·39-s + 0.312·41-s − 0.152·43-s − 1.16·47-s + 9/7·49-s − 0.824·53-s − 0.520·59-s + 0.512·61-s + 0.503·63-s − 0.977·67-s + 0.481·69-s − 0.474·71-s + 1.17·73-s + 1.82·77-s + 0.900·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 206400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 206400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.775979311\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.775979311\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 43 | \( 1 + T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−13.05442533469639, −12.29164613780634, −12.02008815243203, −11.67489840204258, −11.09979590525551, −10.93591561294624, −10.31013260738392, −9.808648508234714, −9.244182731240194, −8.606246554776658, −8.383757773259361, −7.823592674696193, −7.370144713470530, −6.564346462940965, −6.287004098905769, −5.938273007153403, −4.981603161029514, −4.889298597138166, −4.227328458322338, −3.755822767973179, −3.171483614162134, −2.208535484930963, −1.659742199765253, −1.212505107797892, −0.6231098466745421,
0.6231098466745421, 1.212505107797892, 1.659742199765253, 2.208535484930963, 3.171483614162134, 3.755822767973179, 4.227328458322338, 4.889298597138166, 4.981603161029514, 5.938273007153403, 6.287004098905769, 6.564346462940965, 7.370144713470530, 7.823592674696193, 8.383757773259361, 8.606246554776658, 9.244182731240194, 9.808648508234714, 10.31013260738392, 10.93591561294624, 11.09979590525551, 11.67489840204258, 12.02008815243203, 12.29164613780634, 13.05442533469639
