| L(s) = 1 | − 5-s + 6·11-s − 4·13-s − 6·17-s − 4·19-s + 25-s − 6·29-s + 4·31-s − 8·37-s − 8·43-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 − 4·79-s − 12·83-s + 6·85-s + 12·89-s + 4·95-s − 2·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 0.447·5-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 − 1.31·37-s − 1.21·43-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 − 0.450·79-s − 1.31·83-s + 0.650·85-s + 1.27·89-s + 0.410·95-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 141120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 141120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7591525369\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7591525369\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 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
−13.46907213512375, −12.80379913797424, −12.38612716619537, −12.00505790411716, −11.47628529862106, −11.09875840058369, −10.66985693930054, −9.818764059478383, −9.637017318911328, −8.940537753253577, −8.647250216342661, −8.144547432419305, −7.376607757390433, −6.982040405670236, −6.492707470666280, −6.226054697458489, −5.298066493532563, −4.710883367142957, −4.385915610452538, −3.706636737940075, −3.367861630465124, −2.353135804559730, −1.979387880310740, −1.263593550530225, −0.2592872161623165,
0.2592872161623165, 1.263593550530225, 1.979387880310740, 2.353135804559730, 3.367861630465124, 3.706636737940075, 4.385915610452538, 4.710883367142957, 5.298066493532563, 6.226054697458489, 6.492707470666280, 6.982040405670236, 7.376607757390433, 8.144547432419305, 8.647250216342661, 8.940537753253577, 9.637017318911328, 9.818764059478383, 10.66985693930054, 11.09875840058369, 11.47628529862106, 12.00505790411716, 12.38612716619537, 12.80379913797424, 13.46907213512375
