| L(s) = 1 | − 2.10·2-s + 1.69·3-s + 2.44·4-s + 0.492·5-s − 3.58·6-s − 7-s − 0.948·8-s − 0.115·9-s − 1.03·10-s + 4.16·12-s − 5.30·13-s + 2.10·14-s + 0.836·15-s − 2.89·16-s − 3.03·17-s + 0.242·18-s + 4.66·19-s + 1.20·20-s − 1.69·21-s − 5.63·23-s − 1.61·24-s − 4.75·25-s + 11.1·26-s − 5.29·27-s − 2.44·28-s − 6.92·29-s − 1.76·30-s + ⋯ |
| L(s) = 1 | − 1.49·2-s + 0.980·3-s + 1.22·4-s + 0.220·5-s − 1.46·6-s − 0.377·7-s − 0.335·8-s − 0.0383·9-s − 0.328·10-s + 1.20·12-s − 1.47·13-s + 0.563·14-s + 0.215·15-s − 0.724·16-s − 0.736·17-s + 0.0572·18-s + 1.07·19-s + 0.269·20-s − 0.370·21-s − 1.17·23-s − 0.328·24-s − 0.951·25-s + 2.19·26-s − 1.01·27-s − 0.462·28-s − 1.28·29-s − 0.322·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 2.10T + 2T^{2} \) |
| 3 | \( 1 - 1.69T + 3T^{2} \) |
| 5 | \( 1 - 0.492T + 5T^{2} \) |
| 13 | \( 1 + 5.30T + 13T^{2} \) |
| 17 | \( 1 + 3.03T + 17T^{2} \) |
| 19 | \( 1 - 4.66T + 19T^{2} \) |
| 23 | \( 1 + 5.63T + 23T^{2} \) |
| 29 | \( 1 + 6.92T + 29T^{2} \) |
| 31 | \( 1 + 1.26T + 31T^{2} \) |
| 37 | \( 1 - 10.8T + 37T^{2} \) |
| 41 | \( 1 - 1.44T + 41T^{2} \) |
| 43 | \( 1 + 2.88T + 43T^{2} \) |
| 47 | \( 1 + 8.75T + 47T^{2} \) |
| 53 | \( 1 - 6.63T + 53T^{2} \) |
| 59 | \( 1 + 8.35T + 59T^{2} \) |
| 61 | \( 1 - 13.8T + 61T^{2} \) |
| 67 | \( 1 + 9.70T + 67T^{2} \) |
| 71 | \( 1 - 5.94T + 71T^{2} \) |
| 73 | \( 1 - 3.77T + 73T^{2} \) |
| 79 | \( 1 - 8.80T + 79T^{2} \) |
| 83 | \( 1 + 11.0T + 83T^{2} \) |
| 89 | \( 1 - 3.10T + 89T^{2} \) |
| 97 | \( 1 + 6.31T + 97T^{2} \) |
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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
−9.544130453732529018968816312595, −9.193469798443079992831962300415, −8.007219023258487007006194492454, −7.72355166939768481030626212322, −6.74418658139774685707430342003, −5.51425236850674409658562045761, −4.07537894568469433886434490736, −2.70715440237354428203622246974, −1.93084857757062747794727626976, 0,
1.93084857757062747794727626976, 2.70715440237354428203622246974, 4.07537894568469433886434490736, 5.51425236850674409658562045761, 6.74418658139774685707430342003, 7.72355166939768481030626212322, 8.007219023258487007006194492454, 9.193469798443079992831962300415, 9.544130453732529018968816312595
