| L(s) = 1 | − 5-s − 2·7-s + 6·11-s − 4·13-s + 6·17-s + 4·19-s + 25-s + 6·29-s + 4·31-s + 2·35-s + 8·37-s − 8·43-s − 3·49-s + 6·53-s − 6·55-s + 6·59-s + 2·61-s + 4·65-s + 4·67-s − 12·71-s − 10·73-s − 12·77-s + 4·79-s + 12·83-s − 6·85-s − 12·89-s + 8·91-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.755·7-s + 1.80·11-s − 1.10·13-s + 1.45·17-s + 0.917·19-s + 1/5·25-s + 1.11·29-s + 0.718·31-s + 0.338·35-s + 1.31·37-s − 1.21·43-s − 3/7·49-s + 0.824·53-s − 0.809·55-s + 0.781·59-s + 0.256·61-s + 0.496·65-s + 0.488·67-s − 1.42·71-s − 1.17·73-s − 1.36·77-s + 0.450·79-s + 1.31·83-s − 0.650·85-s − 1.27·89-s + 0.838·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.420244348\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.420244348\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 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
−10.03996527820040651492784230004, −9.760247098264765887005783221465, −8.773698297350472018641936567326, −7.72126509708723736389564020435, −6.90901872656979444052935726709, −6.10474012488799251205190118618, −4.89127133089075044081252672966, −3.80702296297205902361458628051, −2.91539889390888382580520587833, −1.06503795092042453789503502485,
1.06503795092042453789503502485, 2.91539889390888382580520587833, 3.80702296297205902361458628051, 4.89127133089075044081252672966, 6.10474012488799251205190118618, 6.90901872656979444052935726709, 7.72126509708723736389564020435, 8.773698297350472018641936567326, 9.760247098264765887005783221465, 10.03996527820040651492784230004
