| L(s) = 1 | + 195.·3-s + 1.88e3·5-s + 6.68e3·7-s + 1.86e4·9-s + 3.38e4·11-s − 2.85e4·13-s + 3.69e5·15-s − 1.00e5·17-s − 5.28e5·19-s + 1.30e6·21-s − 6.35e5·23-s + 1.60e6·25-s − 2.07e5·27-s − 5.60e6·29-s + 4.13e6·31-s + 6.61e6·33-s + 1.26e7·35-s − 3.49e6·37-s − 5.58e6·39-s + 3.30e7·41-s + 3.58e7·43-s + 3.51e7·45-s + 2.81e7·47-s + 4.39e6·49-s − 1.96e7·51-s − 3.87e7·53-s + 6.38e7·55-s + ⋯ |
| L(s) = 1 | + 1.39·3-s + 1.35·5-s + 1.05·7-s + 0.946·9-s + 0.696·11-s − 0.277·13-s + 1.88·15-s − 0.290·17-s − 0.929·19-s + 1.46·21-s − 0.473·23-s + 0.823·25-s − 0.0750·27-s − 1.47·29-s + 0.804·31-s + 0.971·33-s + 1.42·35-s − 0.306·37-s − 0.386·39-s + 1.82·41-s + 1.59·43-s + 1.27·45-s + 0.841·47-s + 0.108·49-s − 0.405·51-s − 0.674·53-s + 0.940·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 52 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 52 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(4.384635654\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.384635654\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 + 2.85e4T \) |
| good | 3 | \( 1 - 195.T + 1.96e4T^{2} \) |
| 5 | \( 1 - 1.88e3T + 1.95e6T^{2} \) |
| 7 | \( 1 - 6.68e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 3.38e4T + 2.35e9T^{2} \) |
| 17 | \( 1 + 1.00e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 5.28e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 6.35e5T + 1.80e12T^{2} \) |
| 29 | \( 1 + 5.60e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 4.13e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 3.49e6T + 1.29e14T^{2} \) |
| 41 | \( 1 - 3.30e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 3.58e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 2.81e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 3.87e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 7.53e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 1.53e8T + 1.16e16T^{2} \) |
| 67 | \( 1 - 1.69e7T + 2.72e16T^{2} \) |
| 71 | \( 1 - 9.46e7T + 4.58e16T^{2} \) |
| 73 | \( 1 - 6.01e7T + 5.88e16T^{2} \) |
| 79 | \( 1 - 1.39e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 8.05e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 7.35e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 1.43e9T + 7.60e17T^{2} \) |
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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
−13.92961343047920195664895503625, −12.70384662222714747523200580061, −10.99159711819737739567422377710, −9.594107961645015270132896774882, −8.841379486262675485635644128634, −7.63437198009574195875679615040, −5.96015460149442630717308180222, −4.26045686128381588926083914422, −2.45417619904281807417056998116, −1.62138566348986792744643400426,
1.62138566348986792744643400426, 2.45417619904281807417056998116, 4.26045686128381588926083914422, 5.96015460149442630717308180222, 7.63437198009574195875679615040, 8.841379486262675485635644128634, 9.594107961645015270132896774882, 10.99159711819737739567422377710, 12.70384662222714747523200580061, 13.92961343047920195664895503625
