| L(s) = 1 | + 361.·2-s − 32.0·4-s − 1.01e5·5-s − 1.96e5·7-s − 4.74e7·8-s − 3.66e7·10-s − 1.10e9·11-s − 1.30e9·13-s − 7.11e7·14-s − 1.71e10·16-s + 3.49e10·17-s − 1.11e11·19-s + 3.24e6·20-s − 4.00e11·22-s − 3.45e11·23-s − 7.52e11·25-s − 4.73e11·26-s + 6.28e6·28-s + 3.38e12·29-s + 4.07e12·31-s + 3.03e9·32-s + 1.26e13·34-s + 1.99e10·35-s + 1.71e13·37-s − 4.02e13·38-s + 4.81e12·40-s + 4.84e13·41-s + ⋯ |
| L(s) = 1 | + 0.999·2-s − 0.000244·4-s − 0.116·5-s − 0.0128·7-s − 1.00·8-s − 0.116·10-s − 1.55·11-s − 0.444·13-s − 0.0128·14-s − 0.999·16-s + 1.21·17-s − 1.50·19-s + 2.83e−5·20-s − 1.55·22-s − 0.920·23-s − 0.986·25-s − 0.444·26-s + 3.14e−6·28-s + 1.25·29-s + 0.857·31-s + 0.000488·32-s + 1.21·34-s + 0.00149·35-s + 0.802·37-s − 1.50·38-s + 0.116·40-s + 0.947·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 2 | \( 1 - 361.T + 1.31e5T^{2} \) |
| 5 | \( 1 + 1.01e5T + 7.62e11T^{2} \) |
| 7 | \( 1 + 1.96e5T + 2.32e14T^{2} \) |
| 11 | \( 1 + 1.10e9T + 5.05e17T^{2} \) |
| 13 | \( 1 + 1.30e9T + 8.65e18T^{2} \) |
| 17 | \( 1 - 3.49e10T + 8.27e20T^{2} \) |
| 19 | \( 1 + 1.11e11T + 5.48e21T^{2} \) |
| 23 | \( 1 + 3.45e11T + 1.41e23T^{2} \) |
| 29 | \( 1 - 3.38e12T + 7.25e24T^{2} \) |
| 31 | \( 1 - 4.07e12T + 2.25e25T^{2} \) |
| 37 | \( 1 - 1.71e13T + 4.56e26T^{2} \) |
| 41 | \( 1 - 4.84e13T + 2.61e27T^{2} \) |
| 43 | \( 1 + 6.03e13T + 5.87e27T^{2} \) |
| 47 | \( 1 - 1.95e14T + 2.66e28T^{2} \) |
| 53 | \( 1 + 3.02e14T + 2.05e29T^{2} \) |
| 59 | \( 1 - 2.96e14T + 1.27e30T^{2} \) |
| 61 | \( 1 - 7.02e14T + 2.24e30T^{2} \) |
| 67 | \( 1 + 2.18e15T + 1.10e31T^{2} \) |
| 71 | \( 1 + 9.52e15T + 2.96e31T^{2} \) |
| 73 | \( 1 + 1.09e16T + 4.74e31T^{2} \) |
| 79 | \( 1 + 1.60e16T + 1.81e32T^{2} \) |
| 83 | \( 1 - 3.08e16T + 4.21e32T^{2} \) |
| 89 | \( 1 + 3.54e16T + 1.37e33T^{2} \) |
| 97 | \( 1 + 5.46e16T + 5.95e33T^{2} \) |
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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
−15.85947537659305065213433989527, −14.58449543458420276848242202145, −13.25561137106467077185716243682, −12.12477482030216004370621508205, −10.14250724063826086247272492793, −8.077348028145380647221480505878, −5.87324591486602368867981996961, −4.45044005683847321545305174941, −2.71633511119956943211308443294, 0,
2.71633511119956943211308443294, 4.45044005683847321545305174941, 5.87324591486602368867981996961, 8.077348028145380647221480505878, 10.14250724063826086247272492793, 12.12477482030216004370621508205, 13.25561137106467077185716243682, 14.58449543458420276848242202145, 15.85947537659305065213433989527
