| L(s) = 1 | + 5-s − 2·7-s + 13-s + 4·17-s − 4·19-s − 23-s + 25-s + 3·29-s − 31-s − 2·35-s − 8·37-s + 5·41-s − 6·43-s − 9·47-s − 3·49-s − 2·53-s + 65-s + 4·67-s − 3·71-s + 7·73-s + 4·79-s − 8·83-s + 4·85-s + 14·89-s − 2·91-s − 4·95-s − 14·97-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.755·7-s + 0.277·13-s + 0.970·17-s − 0.917·19-s − 0.208·23-s + 1/5·25-s + 0.557·29-s − 0.179·31-s − 0.338·35-s − 1.31·37-s + 0.780·41-s − 0.914·43-s − 1.31·47-s − 3/7·49-s − 0.274·53-s + 0.124·65-s + 0.488·67-s − 0.356·71-s + 0.819·73-s + 0.450·79-s − 0.878·83-s + 0.433·85-s + 1.48·89-s − 0.209·91-s − 0.410·95-s − 1.42·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−7.41828221426730488565657012205, −6.58312615606714474227509842902, −6.24103414973355131926441249927, −5.41411450162870433403656312461, −4.73502907055562708350414375027, −3.72156006813250260796216967867, −3.17490651892526508204222014921, −2.22114910796226392630475232768, −1.29434324689996725627123353605, 0,
1.29434324689996725627123353605, 2.22114910796226392630475232768, 3.17490651892526508204222014921, 3.72156006813250260796216967867, 4.73502907055562708350414375027, 5.41411450162870433403656312461, 6.24103414973355131926441249927, 6.58312615606714474227509842902, 7.41828221426730488565657012205
