| L(s) = 1 | + 500.·2-s + 1.19e5·4-s + 3.90e5·5-s + 8.90e6·7-s − 5.87e6·8-s + 1.95e8·10-s − 4.48e8·11-s − 3.89e9·13-s + 4.45e9·14-s − 1.85e10·16-s + 5.96e8·17-s − 4.36e10·19-s + 4.66e10·20-s − 2.24e11·22-s − 7.23e10·23-s + 1.52e11·25-s − 1.95e12·26-s + 1.06e12·28-s − 1.82e12·29-s + 5.27e12·31-s − 8.52e12·32-s + 2.98e11·34-s + 3.47e12·35-s − 1.62e13·37-s − 2.18e13·38-s − 2.29e12·40-s − 9.78e12·41-s + ⋯ |
| L(s) = 1 | + 1.38·2-s + 0.910·4-s + 0.447·5-s + 0.583·7-s − 0.123·8-s + 0.618·10-s − 0.631·11-s − 1.32·13-s + 0.807·14-s − 1.08·16-s + 0.0207·17-s − 0.590·19-s + 0.407·20-s − 0.872·22-s − 0.192·23-s + 0.200·25-s − 1.83·26-s + 0.531·28-s − 0.675·29-s + 1.11·31-s − 1.37·32-s + 0.0286·34-s + 0.261·35-s − 0.761·37-s − 0.815·38-s − 0.0553·40-s − 0.191·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(9)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{19}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - 3.90e5T \) |
| good | 2 | \( 1 - 500.T + 1.31e5T^{2} \) |
| 7 | \( 1 - 8.90e6T + 2.32e14T^{2} \) |
| 11 | \( 1 + 4.48e8T + 5.05e17T^{2} \) |
| 13 | \( 1 + 3.89e9T + 8.65e18T^{2} \) |
| 17 | \( 1 - 5.96e8T + 8.27e20T^{2} \) |
| 19 | \( 1 + 4.36e10T + 5.48e21T^{2} \) |
| 23 | \( 1 + 7.23e10T + 1.41e23T^{2} \) |
| 29 | \( 1 + 1.82e12T + 7.25e24T^{2} \) |
| 31 | \( 1 - 5.27e12T + 2.25e25T^{2} \) |
| 37 | \( 1 + 1.62e13T + 4.56e26T^{2} \) |
| 41 | \( 1 + 9.78e12T + 2.61e27T^{2} \) |
| 43 | \( 1 - 1.46e14T + 5.87e27T^{2} \) |
| 47 | \( 1 + 2.43e14T + 2.66e28T^{2} \) |
| 53 | \( 1 + 6.84e14T + 2.05e29T^{2} \) |
| 59 | \( 1 + 9.83e14T + 1.27e30T^{2} \) |
| 61 | \( 1 + 1.84e15T + 2.24e30T^{2} \) |
| 67 | \( 1 + 2.46e15T + 1.10e31T^{2} \) |
| 71 | \( 1 + 9.43e14T + 2.96e31T^{2} \) |
| 73 | \( 1 - 1.15e16T + 4.74e31T^{2} \) |
| 79 | \( 1 - 1.58e16T + 1.81e32T^{2} \) |
| 83 | \( 1 + 1.62e16T + 4.21e32T^{2} \) |
| 89 | \( 1 - 4.03e16T + 1.37e33T^{2} \) |
| 97 | \( 1 + 7.85e16T + 5.95e33T^{2} \) |
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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
−12.06637390419443477240562260162, −10.79974113491880739172605522028, −9.427521008066590571667480347610, −7.83526654679336641580376081499, −6.41431970857116435352113833570, −5.22358436313140554860731542348, −4.49541944877359502726347362713, −2.96699917377600389050225979290, −1.95291889765795639343791989593, 0,
1.95291889765795639343791989593, 2.96699917377600389050225979290, 4.49541944877359502726347362713, 5.22358436313140554860731542348, 6.41431970857116435352113833570, 7.83526654679336641580376081499, 9.427521008066590571667480347610, 10.79974113491880739172605522028, 12.06637390419443477240562260162
