| L(s) = 1 | + 5.08·2-s − 6.16·4-s − 41.8·5-s − 193.·8-s − 212.·10-s − 72.0·11-s − 632.·13-s − 788.·16-s + 1.97e3·17-s − 1.86e3·19-s + 257.·20-s − 366.·22-s − 413.·23-s − 1.37e3·25-s − 3.21e3·26-s − 731.·29-s + 6.12e3·31-s + 2.19e3·32-s + 1.00e4·34-s + 1.03e4·37-s − 9.47e3·38-s + 8.11e3·40-s + 3.52e3·41-s − 1.45e4·43-s + 444.·44-s − 2.10e3·46-s + 2.14e4·47-s + ⋯ |
| L(s) = 1 | + 0.898·2-s − 0.192·4-s − 0.748·5-s − 1.07·8-s − 0.672·10-s − 0.179·11-s − 1.03·13-s − 0.770·16-s + 1.65·17-s − 1.18·19-s + 0.144·20-s − 0.161·22-s − 0.163·23-s − 0.440·25-s − 0.932·26-s − 0.161·29-s + 1.14·31-s + 0.379·32-s + 1.48·34-s + 1.24·37-s − 1.06·38-s + 0.801·40-s + 0.327·41-s − 1.19·43-s + 0.0346·44-s − 0.146·46-s + 1.41·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.713424692\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.713424692\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 5.08T + 32T^{2} \) |
| 5 | \( 1 + 41.8T + 3.12e3T^{2} \) |
| 11 | \( 1 + 72.0T + 1.61e5T^{2} \) |
| 13 | \( 1 + 632.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.97e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.86e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 413.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 731.T + 2.05e7T^{2} \) |
| 31 | \( 1 - 6.12e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.03e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 3.52e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.45e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.14e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.25e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.61e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 4.02e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.55e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 1.21e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 1.95e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 3.60e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 2.45e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 7.02e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.05e5T + 8.58e9T^{2} \) |
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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
−10.28889319015732535972350739935, −9.513754762161066758765877049684, −8.316608106660271744799049174614, −7.61921877682417396642082379618, −6.35743810614738602675630156248, −5.35812864255038493127223488734, −4.45467313845485669988795822377, −3.61484826467904793548486344018, −2.50824676302521373147626318055, −0.57519829511452390250946893609,
0.57519829511452390250946893609, 2.50824676302521373147626318055, 3.61484826467904793548486344018, 4.45467313845485669988795822377, 5.35812864255038493127223488734, 6.35743810614738602675630156248, 7.61921877682417396642082379618, 8.316608106660271744799049174614, 9.513754762161066758765877049684, 10.28889319015732535972350739935
