L(s) = 1 | + 5-s + 4·11-s − 2·13-s − 2·17-s + 4·23-s + 25-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 + 4·55-s + 12·59-s − 2·61-s − 2·65-s + 4·67-s − 8·71-s − 6·73-s − 4·79-s − 12·83-s − 2·85-s + 10·89-s − 2·97-s + 101-s + ⋯ |
L(s) = 1 | + 0.447·5-s + 1.20·11-s − 0.554·13-s − 0.485·17-s + 0.834·23-s + 1/5·25-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 + 0.539·55-s + 1.56·59-s − 0.256·61-s − 0.248·65-s + 0.488·67-s − 0.949·71-s − 0.702·73-s − 0.450·79-s − 1.31·83-s − 0.216·85-s + 1.05·89-s − 0.203·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 259920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 259920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 19 | \( 1 \) |
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
−13.15263265755023, −12.58825886500875, −12.14846800703142, −11.51948686890913, −11.38291029130956, −10.77582665913373, −10.15859943645948, −9.778819946322279, −9.407422849965750, −8.863728251477963, −8.512360371138813, −7.920524583592756, −7.323255916911883, −6.851567785415544, −6.396515229109145, −6.079556579797263, −5.380406940144754, −4.825405149004886, −4.373264283716387, −3.945365669341379, −3.145775951450802, −2.646863490270586, −2.201318511479082, −1.235106739860111, −1.051757011918343, 0,
1.051757011918343, 1.235106739860111, 2.201318511479082, 2.646863490270586, 3.145775951450802, 3.945365669341379, 4.373264283716387, 4.825405149004886, 5.380406940144754, 6.079556579797263, 6.396515229109145, 6.851567785415544, 7.323255916911883, 7.920524583592756, 8.512360371138813, 8.863728251477963, 9.407422849965750, 9.778819946322279, 10.15859943645948, 10.77582665913373, 11.38291029130956, 11.51948686890913, 12.14846800703142, 12.58825886500875, 13.15263265755023