| L(s) = 1 | − 5-s − 6·13-s − 2·17-s − 2·19-s − 6·23-s + 25-s − 6·29-s − 4·31-s − 8·37-s − 8·41-s − 43-s − 6·47-s − 7·49-s − 6·53-s − 4·59-s − 14·61-s + 6·65-s + 4·67-s + 8·71-s − 4·73-s + 12·79-s + 2·83-s + 2·85-s + 14·89-s + 2·95-s − 2·97-s + 101-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.66·13-s − 0.485·17-s − 0.458·19-s − 1.25·23-s + 1/5·25-s − 1.11·29-s − 0.718·31-s − 1.31·37-s − 1.24·41-s − 0.152·43-s − 0.875·47-s − 49-s − 0.824·53-s − 0.520·59-s − 1.79·61-s + 0.744·65-s + 0.488·67-s + 0.949·71-s − 0.468·73-s + 1.35·79-s + 0.219·83-s + 0.216·85-s + 1.48·89-s + 0.205·95-s − 0.203·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 43 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 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
−15.54872909389621, −15.08250775886423, −14.61864422567251, −14.15543698451094, −13.50071795787453, −12.92990346177129, −12.31600942827714, −12.04245506204743, −11.43650711336482, −10.77279815624374, −10.34117093177609, −9.606757418974514, −9.291048100601787, −8.538710640063080, −7.797901428708296, −7.644210830210262, −6.714292760863829, −6.463794931790917, −5.420178627488385, −5.010172241837863, −4.374285942599860, −3.664280982197907, −3.060312321667707, −2.096430531278342, −1.704893012055578, 0, 0,
1.704893012055578, 2.096430531278342, 3.060312321667707, 3.664280982197907, 4.374285942599860, 5.010172241837863, 5.420178627488385, 6.463794931790917, 6.714292760863829, 7.644210830210262, 7.797901428708296, 8.538710640063080, 9.291048100601787, 9.606757418974514, 10.34117093177609, 10.77279815624374, 11.43650711336482, 12.04245506204743, 12.31600942827714, 12.92990346177129, 13.50071795787453, 14.15543698451094, 14.61864422567251, 15.08250775886423, 15.54872909389621
