| L(s) = 1 | + 0.478·2-s − 1.77·4-s − 3.51·5-s + 0.805·7-s − 1.80·8-s − 1.68·10-s − 3.54·11-s + 6.61·13-s + 0.385·14-s + 2.67·16-s + 0.933·17-s − 3.74·19-s + 6.22·20-s − 1.69·22-s + 7.34·25-s + 3.16·26-s − 1.42·28-s − 1.58·29-s + 0.565·31-s + 4.89·32-s + 0.447·34-s − 2.83·35-s + 8.17·37-s − 1.79·38-s + 6.34·40-s + 3.81·41-s − 6.85·43-s + ⋯ |
| L(s) = 1 | + 0.338·2-s − 0.885·4-s − 1.57·5-s + 0.304·7-s − 0.638·8-s − 0.532·10-s − 1.06·11-s + 1.83·13-s + 0.103·14-s + 0.669·16-s + 0.226·17-s − 0.860·19-s + 1.39·20-s − 0.362·22-s + 1.46·25-s + 0.620·26-s − 0.269·28-s − 0.293·29-s + 0.101·31-s + 0.865·32-s + 0.0767·34-s − 0.478·35-s + 1.34·37-s − 0.291·38-s + 1.00·40-s + 0.595·41-s − 1.04·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4761 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4761 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 23 | \( 1 \) |
| good | 2 | \( 1 - 0.478T + 2T^{2} \) |
| 5 | \( 1 + 3.51T + 5T^{2} \) |
| 7 | \( 1 - 0.805T + 7T^{2} \) |
| 11 | \( 1 + 3.54T + 11T^{2} \) |
| 13 | \( 1 - 6.61T + 13T^{2} \) |
| 17 | \( 1 - 0.933T + 17T^{2} \) |
| 19 | \( 1 + 3.74T + 19T^{2} \) |
| 29 | \( 1 + 1.58T + 29T^{2} \) |
| 31 | \( 1 - 0.565T + 31T^{2} \) |
| 37 | \( 1 - 8.17T + 37T^{2} \) |
| 41 | \( 1 - 3.81T + 41T^{2} \) |
| 43 | \( 1 + 6.85T + 43T^{2} \) |
| 47 | \( 1 - 5.69T + 47T^{2} \) |
| 53 | \( 1 - 6.98T + 53T^{2} \) |
| 59 | \( 1 + 1.01T + 59T^{2} \) |
| 61 | \( 1 - 12.2T + 61T^{2} \) |
| 67 | \( 1 + 2.48T + 67T^{2} \) |
| 71 | \( 1 + 13.2T + 71T^{2} \) |
| 73 | \( 1 - 6.70T + 73T^{2} \) |
| 79 | \( 1 + 1.43T + 79T^{2} \) |
| 83 | \( 1 + 1.75T + 83T^{2} \) |
| 89 | \( 1 + 4.68T + 89T^{2} \) |
| 97 | \( 1 + 0.794T + 97T^{2} \) |
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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
−8.146418166907085635831986734784, −7.43283635338014408921919409621, −6.36727664723055678147361045503, −5.61609989233622272014475514290, −4.79719646849326018736468751516, −4.05010394506832405647700244299, −3.68020633439857919666692796704, −2.72185534053729024439553034607, −1.07719848092857177778023866955, 0,
1.07719848092857177778023866955, 2.72185534053729024439553034607, 3.68020633439857919666692796704, 4.05010394506832405647700244299, 4.79719646849326018736468751516, 5.61609989233622272014475514290, 6.36727664723055678147361045503, 7.43283635338014408921919409621, 8.146418166907085635831986734784
