| L(s) = 1 | + 3·5-s − 4·7-s + 13-s − 3·17-s + 4·19-s + 4·25-s − 9·29-s − 4·31-s − 12·35-s + 37-s + 6·41-s − 8·43-s − 12·47-s + 9·49-s + 6·53-s + 61-s + 3·65-s + 4·67-s − 12·71-s + 11·73-s − 16·79-s + 12·83-s − 9·85-s − 3·89-s − 4·91-s + 12·95-s + 2·97-s + ⋯ |
| L(s) = 1 | + 1.34·5-s − 1.51·7-s + 0.277·13-s − 0.727·17-s + 0.917·19-s + 4/5·25-s − 1.67·29-s − 0.718·31-s − 2.02·35-s + 0.164·37-s + 0.937·41-s − 1.21·43-s − 1.75·47-s + 9/7·49-s + 0.824·53-s + 0.128·61-s + 0.372·65-s + 0.488·67-s − 1.42·71-s + 1.28·73-s − 1.80·79-s + 1.31·83-s − 0.976·85-s − 0.317·89-s − 0.419·91-s + 1.23·95-s + 0.203·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 5 | \( 1 - 3 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 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
−7.77133374356300284680128512104, −6.89070553337933054938428171633, −6.42718799904595728230962453845, −5.72247365022694368090686428400, −5.20981735085530076681492595480, −3.96148093445634060055559370451, −3.22684654073668095375893389695, −2.39288338822089294054098347965, −1.47843412021064514807145846869, 0,
1.47843412021064514807145846869, 2.39288338822089294054098347965, 3.22684654073668095375893389695, 3.96148093445634060055559370451, 5.20981735085530076681492595480, 5.72247365022694368090686428400, 6.42718799904595728230962453845, 6.89070553337933054938428171633, 7.77133374356300284680128512104
