| L(s) = 1 | − 5.29·5-s − 68.7·11-s + 30·13-s − 121.·17-s + 100·19-s + 89.9·23-s − 97·25-s − 232.·29-s + 180·31-s − 118·37-s + 15.8·41-s − 412·43-s − 285.·47-s + 285.·53-s + 364·55-s + 836.·59-s + 378·61-s − 158.·65-s + 244·67-s + 439.·71-s − 670·73-s + 216·79-s − 804.·83-s + 644·85-s + 968.·89-s − 529.·95-s − 574·97-s + ⋯ |
| L(s) = 1 | − 0.473·5-s − 1.88·11-s + 0.640·13-s − 1.73·17-s + 1.20·19-s + 0.815·23-s − 0.776·25-s − 1.49·29-s + 1.04·31-s − 0.524·37-s + 0.0604·41-s − 1.46·43-s − 0.886·47-s + 0.740·53-s + 0.892·55-s + 1.84·59-s + 0.793·61-s − 0.302·65-s + 0.444·67-s + 0.734·71-s − 1.07·73-s + 0.307·79-s − 1.06·83-s + 0.821·85-s + 1.15·89-s − 0.571·95-s − 0.600·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.074380807\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.074380807\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 5.29T + 125T^{2} \) |
| 11 | \( 1 + 68.7T + 1.33e3T^{2} \) |
| 13 | \( 1 - 30T + 2.19e3T^{2} \) |
| 17 | \( 1 + 121.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 100T + 6.85e3T^{2} \) |
| 23 | \( 1 - 89.9T + 1.21e4T^{2} \) |
| 29 | \( 1 + 232.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 180T + 2.97e4T^{2} \) |
| 37 | \( 1 + 118T + 5.06e4T^{2} \) |
| 41 | \( 1 - 15.8T + 6.89e4T^{2} \) |
| 43 | \( 1 + 412T + 7.95e4T^{2} \) |
| 47 | \( 1 + 285.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 285.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 836.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 378T + 2.26e5T^{2} \) |
| 67 | \( 1 - 244T + 3.00e5T^{2} \) |
| 71 | \( 1 - 439.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 670T + 3.89e5T^{2} \) |
| 79 | \( 1 - 216T + 4.93e5T^{2} \) |
| 83 | \( 1 + 804.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 968.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 574T + 9.12e5T^{2} \) |
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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
−8.772040376917229081863547340256, −8.166122226495908955299271776085, −7.41360526545243904603173065320, −6.67267951142334252459173641316, −5.52407743828611152620850820919, −4.95977364740395436318565558672, −3.88459478477366675155580060874, −2.95236445577973634274193445283, −1.98069997311006881010550352439, −0.46859321321541673835485167941,
0.46859321321541673835485167941, 1.98069997311006881010550352439, 2.95236445577973634274193445283, 3.88459478477366675155580060874, 4.95977364740395436318565558672, 5.52407743828611152620850820919, 6.67267951142334252459173641316, 7.41360526545243904603173065320, 8.166122226495908955299271776085, 8.772040376917229081863547340256
