| L(s) = 1 | − 26·3-s + 64·4-s − 125·5-s + 343·7-s − 53·9-s + 2.52e3·11-s − 1.66e3·12-s + 2.77e3·13-s + 3.25e3·15-s + 4.09e3·16-s − 754·17-s − 8.00e3·20-s − 8.91e3·21-s + 1.56e4·25-s + 2.03e4·27-s + 2.19e4·28-s − 4.58e4·29-s − 6.55e4·33-s − 4.28e4·35-s − 3.39e3·36-s − 7.21e4·39-s + 1.61e5·44-s + 6.62e3·45-s + 1.75e5·47-s − 1.06e5·48-s + 1.17e5·49-s + 1.96e4·51-s + ⋯ |
| L(s) = 1 | − 0.962·3-s + 4-s − 5-s + 7-s − 0.0727·9-s + 1.89·11-s − 0.962·12-s + 1.26·13-s + 0.962·15-s + 16-s − 0.153·17-s − 20-s − 0.962·21-s + 25-s + 1.03·27-s + 28-s − 1.88·29-s − 1.82·33-s − 35-s − 0.0727·36-s − 1.21·39-s + 1.89·44-s + 0.0727·45-s + 1.69·47-s − 0.962·48-s + 49-s + 0.147·51-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.533629180\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.533629180\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + p^{3} T \) |
| 7 | \( 1 - p^{3} T \) |
| good | 2 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 3 | \( 1 + 26 T + p^{6} T^{2} \) |
| 11 | \( 1 - 2522 T + p^{6} T^{2} \) |
| 13 | \( 1 - 2774 T + p^{6} T^{2} \) |
| 17 | \( 1 + 754 T + p^{6} T^{2} \) |
| 19 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 23 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 29 | \( 1 + 45862 T + p^{6} T^{2} \) |
| 31 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 37 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 41 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 43 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 47 | \( 1 - 175646 T + p^{6} T^{2} \) |
| 53 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 59 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 61 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 67 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 71 | \( 1 + 30238 T + p^{6} T^{2} \) |
| 73 | \( 1 - 504254 T + p^{6} T^{2} \) |
| 79 | \( 1 + 930382 T + p^{6} T^{2} \) |
| 83 | \( 1 + 1141306 T + p^{6} T^{2} \) |
| 89 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 97 | \( 1 + 897874 T + p^{6} T^{2} \) |
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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
−15.38418510639657070571684167612, −14.37085498980928211670276930877, −12.22155281071409374069122618549, −11.34599715867335853829387917638, −11.06960231588020210960312930559, −8.639704737298813270913666518956, −7.12613176545687292716744732919, −5.87982423106227430820834566072, −3.92590197416176360634314778117, −1.23913877545426626702327046161,
1.23913877545426626702327046161, 3.92590197416176360634314778117, 5.87982423106227430820834566072, 7.12613176545687292716744732919, 8.639704737298813270913666518956, 11.06960231588020210960312930559, 11.34599715867335853829387917638, 12.22155281071409374069122618549, 14.37085498980928211670276930877, 15.38418510639657070571684167612
