| L(s) = 1 | + 2-s + 1.39·3-s + 4-s + 2.16·5-s + 1.39·6-s − 1.18·7-s + 8-s − 1.05·9-s + 2.16·10-s + 1.39·12-s + 6.23·13-s − 1.18·14-s + 3.01·15-s + 16-s + 2.02·17-s − 1.05·18-s + 19-s + 2.16·20-s − 1.64·21-s − 3.76·23-s + 1.39·24-s − 0.331·25-s + 6.23·26-s − 5.65·27-s − 1.18·28-s + 3.46·29-s + 3.01·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.805·3-s + 0.5·4-s + 0.966·5-s + 0.569·6-s − 0.446·7-s + 0.353·8-s − 0.351·9-s + 0.683·10-s + 0.402·12-s + 1.73·13-s − 0.315·14-s + 0.778·15-s + 0.250·16-s + 0.491·17-s − 0.248·18-s + 0.229·19-s + 0.483·20-s − 0.359·21-s − 0.785·23-s + 0.284·24-s − 0.0662·25-s + 1.22·26-s − 1.08·27-s − 0.223·28-s + 0.644·29-s + 0.550·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.087223532\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.087223532\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 11 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 - 1.39T + 3T^{2} \) |
| 5 | \( 1 - 2.16T + 5T^{2} \) |
| 7 | \( 1 + 1.18T + 7T^{2} \) |
| 13 | \( 1 - 6.23T + 13T^{2} \) |
| 17 | \( 1 - 2.02T + 17T^{2} \) |
| 23 | \( 1 + 3.76T + 23T^{2} \) |
| 29 | \( 1 - 3.46T + 29T^{2} \) |
| 31 | \( 1 - 8.06T + 31T^{2} \) |
| 37 | \( 1 - 5.03T + 37T^{2} \) |
| 41 | \( 1 + 2.39T + 41T^{2} \) |
| 43 | \( 1 - 7.95T + 43T^{2} \) |
| 47 | \( 1 - 3.04T + 47T^{2} \) |
| 53 | \( 1 + 1.12T + 53T^{2} \) |
| 59 | \( 1 + 2.44T + 59T^{2} \) |
| 61 | \( 1 + 11.2T + 61T^{2} \) |
| 67 | \( 1 + 10.1T + 67T^{2} \) |
| 71 | \( 1 - 0.580T + 71T^{2} \) |
| 73 | \( 1 - 8.25T + 73T^{2} \) |
| 79 | \( 1 - 6.77T + 79T^{2} \) |
| 83 | \( 1 - 17.7T + 83T^{2} \) |
| 89 | \( 1 - 5.72T + 89T^{2} \) |
| 97 | \( 1 + 2.56T + 97T^{2} \) |
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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
−8.195169150535127741576897472005, −7.79116815287236613109684955302, −6.48688250492237087131215842099, −6.12804170293541672066562975287, −5.58661026867072627063912250713, −4.47414971459272135488988835717, −3.60547181869162787690512300534, −2.99053074385390050089197714623, −2.19596367315335952905799158654, −1.17622427641069316580397458379,
1.17622427641069316580397458379, 2.19596367315335952905799158654, 2.99053074385390050089197714623, 3.60547181869162787690512300534, 4.47414971459272135488988835717, 5.58661026867072627063912250713, 6.12804170293541672066562975287, 6.48688250492237087131215842099, 7.79116815287236613109684955302, 8.195169150535127741576897472005
