| L(s) = 1 | − 4·11-s + 2·13-s + 2·17-s − 19-s − 4·23-s + 6·29-s − 4·31-s − 6·37-s − 10·41-s + 4·43-s + 12·47-s − 7·49-s − 6·53-s + 12·59-s + 2·61-s − 4·67-s + 8·71-s + 6·73-s + 4·79-s − 12·83-s − 10·89-s − 2·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 1.20·11-s + 0.554·13-s + 0.485·17-s − 0.229·19-s − 0.834·23-s + 1.11·29-s − 0.718·31-s − 0.986·37-s − 1.56·41-s + 0.609·43-s + 1.75·47-s − 49-s − 0.824·53-s + 1.56·59-s + 0.256·61-s − 0.488·67-s + 0.949·71-s + 0.702·73-s + 0.450·79-s − 1.31·83-s − 1.05·89-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 273600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−13.01291574593316, −12.51622937035768, −12.12748936290825, −11.70570833042958, −10.99577107785509, −10.75558120523419, −10.20845628312688, −9.921466267162091, −9.348453170054466, −8.669963018053067, −8.315251017063653, −8.019690674808508, −7.350248066682378, −6.953427112182894, −6.368002199590307, −5.859090519147186, −5.283124003889653, −5.070307909568357, −4.262567161903138, −3.809861360046974, −3.225566881808338, −2.691722918248576, −2.085968525446568, −1.515468989305711, −0.7078118080168380, 0,
0.7078118080168380, 1.515468989305711, 2.085968525446568, 2.691722918248576, 3.225566881808338, 3.809861360046974, 4.262567161903138, 5.070307909568357, 5.283124003889653, 5.859090519147186, 6.368002199590307, 6.953427112182894, 7.350248066682378, 8.019690674808508, 8.315251017063653, 8.669963018053067, 9.348453170054466, 9.921466267162091, 10.20845628312688, 10.75558120523419, 10.99577107785509, 11.70570833042958, 12.12748936290825, 12.51622937035768, 13.01291574593316
