| L(s) = 1 | − 2.60·2-s + 0.980·3-s + 4.77·4-s − 1.69·5-s − 2.55·6-s − 1.30·7-s − 7.23·8-s − 2.03·9-s + 4.40·10-s − 3.27·11-s + 4.68·12-s + 4.30·13-s + 3.39·14-s − 1.65·15-s + 9.27·16-s − 1.02·17-s + 5.31·18-s − 7.01·19-s − 8.08·20-s − 1.27·21-s + 8.51·22-s + 0.835·23-s − 7.09·24-s − 2.13·25-s − 11.2·26-s − 4.93·27-s − 6.24·28-s + ⋯ |
| L(s) = 1 | − 1.84·2-s + 0.565·3-s + 2.38·4-s − 0.756·5-s − 1.04·6-s − 0.493·7-s − 2.55·8-s − 0.679·9-s + 1.39·10-s − 0.986·11-s + 1.35·12-s + 1.19·13-s + 0.908·14-s − 0.427·15-s + 2.31·16-s − 0.248·17-s + 1.25·18-s − 1.60·19-s − 1.80·20-s − 0.279·21-s + 1.81·22-s + 0.174·23-s − 1.44·24-s − 0.427·25-s − 2.19·26-s − 0.950·27-s − 1.17·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 241 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 241 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 241 | \( 1 + T \) |
| good | 2 | \( 1 + 2.60T + 2T^{2} \) |
| 3 | \( 1 - 0.980T + 3T^{2} \) |
| 5 | \( 1 + 1.69T + 5T^{2} \) |
| 7 | \( 1 + 1.30T + 7T^{2} \) |
| 11 | \( 1 + 3.27T + 11T^{2} \) |
| 13 | \( 1 - 4.30T + 13T^{2} \) |
| 17 | \( 1 + 1.02T + 17T^{2} \) |
| 19 | \( 1 + 7.01T + 19T^{2} \) |
| 23 | \( 1 - 0.835T + 23T^{2} \) |
| 29 | \( 1 + 1.11T + 29T^{2} \) |
| 31 | \( 1 + 3.97T + 31T^{2} \) |
| 37 | \( 1 + 11.3T + 37T^{2} \) |
| 41 | \( 1 - 1.22T + 41T^{2} \) |
| 43 | \( 1 - 10.8T + 43T^{2} \) |
| 47 | \( 1 + 0.151T + 47T^{2} \) |
| 53 | \( 1 - 3.02T + 53T^{2} \) |
| 59 | \( 1 + 4.15T + 59T^{2} \) |
| 61 | \( 1 - 5.62T + 61T^{2} \) |
| 67 | \( 1 - 12.9T + 67T^{2} \) |
| 71 | \( 1 + 11.2T + 71T^{2} \) |
| 73 | \( 1 - 11.7T + 73T^{2} \) |
| 79 | \( 1 + 1.66T + 79T^{2} \) |
| 83 | \( 1 + 2.34T + 83T^{2} \) |
| 89 | \( 1 - 18.1T + 89T^{2} \) |
| 97 | \( 1 + 7.17T + 97T^{2} \) |
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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
−11.03835023411756180973500029911, −10.74011048203502500799704413781, −9.451222013953380327854596580950, −8.549075689020562296361643688747, −8.160936927145723422513195189383, −7.08729919571522677385995916133, −5.94676966412518300134568301165, −3.57888668523716984072157735305, −2.25453857323099446533056531089, 0,
2.25453857323099446533056531089, 3.57888668523716984072157735305, 5.94676966412518300134568301165, 7.08729919571522677385995916133, 8.160936927145723422513195189383, 8.549075689020562296361643688747, 9.451222013953380327854596580950, 10.74011048203502500799704413781, 11.03835023411756180973500029911
