| L(s) = 1 | + 8.11·5-s − 11·11-s + 8.35·23-s + 40.8·25-s + 24.5·31-s + 72.8·37-s + 50·47-s + 49·49-s + 70·53-s − 89.2·55-s − 96.5·59-s + 129.·67-s + 23.4·71-s − 177.·89-s − 193.·97-s + 190·103-s + 47.1·113-s + 67.7·115-s + ⋯ |
| L(s) = 1 | + 1.62·5-s − 11-s + 0.363·23-s + 1.63·25-s + 0.793·31-s + 1.96·37-s + 1.06·47-s + 0.999·49-s + 1.32·53-s − 1.62·55-s − 1.63·59-s + 1.93·67-s + 0.329·71-s − 1.99·89-s − 1.99·97-s + 1.84·103-s + 0.417·113-s + 0.589·115-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1584 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.736581120\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.736581120\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 + 11T \) |
| good | 5 | \( 1 - 8.11T + 25T^{2} \) |
| 7 | \( 1 - 49T^{2} \) |
| 13 | \( 1 - 169T^{2} \) |
| 17 | \( 1 - 289T^{2} \) |
| 19 | \( 1 - 361T^{2} \) |
| 23 | \( 1 - 8.35T + 529T^{2} \) |
| 29 | \( 1 - 841T^{2} \) |
| 31 | \( 1 - 24.5T + 961T^{2} \) |
| 37 | \( 1 - 72.8T + 1.36e3T^{2} \) |
| 41 | \( 1 - 1.68e3T^{2} \) |
| 43 | \( 1 - 1.84e3T^{2} \) |
| 47 | \( 1 - 50T + 2.20e3T^{2} \) |
| 53 | \( 1 - 70T + 2.80e3T^{2} \) |
| 59 | \( 1 + 96.5T + 3.48e3T^{2} \) |
| 61 | \( 1 - 3.72e3T^{2} \) |
| 67 | \( 1 - 129.T + 4.48e3T^{2} \) |
| 71 | \( 1 - 23.4T + 5.04e3T^{2} \) |
| 73 | \( 1 - 5.32e3T^{2} \) |
| 79 | \( 1 - 6.24e3T^{2} \) |
| 83 | \( 1 - 6.88e3T^{2} \) |
| 89 | \( 1 + 177.T + 7.92e3T^{2} \) |
| 97 | \( 1 + 193.T + 9.40e3T^{2} \) |
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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
−9.379925407138494022336187500733, −8.546743388893875586090151511044, −7.63546658731444965585532985166, −6.68666641823714000335451223329, −5.85791914706453227652528984263, −5.32961407865246960548285954206, −4.33545581512937863458919565480, −2.83711206751063116636007782584, −2.23426977340438872556270924408, −0.954778021733146979436581372230,
0.954778021733146979436581372230, 2.23426977340438872556270924408, 2.83711206751063116636007782584, 4.33545581512937863458919565480, 5.32961407865246960548285954206, 5.85791914706453227652528984263, 6.68666641823714000335451223329, 7.63546658731444965585532985166, 8.546743388893875586090151511044, 9.379925407138494022336187500733
