| L(s) = 1 | − 10.6·2-s − 9·3-s + 80.6·4-s + 25·5-s + 95.5·6-s + 105.·7-s − 515.·8-s + 81·9-s − 265.·10-s + 447.·11-s − 725.·12-s + 276.·13-s − 1.12e3·14-s − 225·15-s + 2.89e3·16-s + 1.82e3·17-s − 859.·18-s − 1.37e3·19-s + 2.01e3·20-s − 952.·21-s − 4.74e3·22-s − 1.12e3·23-s + 4.64e3·24-s + 625·25-s − 2.93e3·26-s − 729·27-s + 8.52e3·28-s + ⋯ |
| L(s) = 1 | − 1.87·2-s − 0.577·3-s + 2.51·4-s + 0.447·5-s + 1.08·6-s + 0.816·7-s − 2.84·8-s + 0.333·9-s − 0.838·10-s + 1.11·11-s − 1.45·12-s + 0.453·13-s − 1.53·14-s − 0.258·15-s + 2.82·16-s + 1.53·17-s − 0.625·18-s − 0.871·19-s + 1.12·20-s − 0.471·21-s − 2.09·22-s − 0.442·23-s + 1.64·24-s + 0.200·25-s − 0.850·26-s − 0.192·27-s + 2.05·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.5983703157\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5983703157\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 9T \) |
| 5 | \( 1 - 25T \) |
| good | 2 | \( 1 + 10.6T + 32T^{2} \) |
| 7 | \( 1 - 105.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 447.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 276.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.82e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.37e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 1.12e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 1.62e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 443.T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.25e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.68e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 8.86e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.77e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.01e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.31e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.59e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.93e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 5.28e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 3.57e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 9.18e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 2.02e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.26e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.38e5T + 8.58e9T^{2} \) |
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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
−18.12263773202914897961246092323, −17.20920571914792901841949142862, −16.40483348009354710859080364461, −14.69342112070556311205848048584, −11.98695279794919736999404098474, −10.83794874819492548125667921090, −9.538055987989469830038023241659, −8.027636115051473437793805512960, −6.29825216198372449813050528411, −1.35969079090521041025650949082,
1.35969079090521041025650949082, 6.29825216198372449813050528411, 8.027636115051473437793805512960, 9.538055987989469830038023241659, 10.83794874819492548125667921090, 11.98695279794919736999404098474, 14.69342112070556311205848048584, 16.40483348009354710859080364461, 17.20920571914792901841949142862, 18.12263773202914897961246092323
