| L(s) = 1 | + 0.497·2-s − 3-s − 1.75·4-s − 5-s − 0.497·6-s − 2.87·7-s − 1.86·8-s + 9-s − 0.497·10-s − 2.59·11-s + 1.75·12-s − 6.61·13-s − 1.42·14-s + 15-s + 2.57·16-s + 0.503·17-s + 0.497·18-s + 5.19·19-s + 1.75·20-s + 2.87·21-s − 1.28·22-s + 1.86·24-s + 25-s − 3.29·26-s − 27-s + 5.03·28-s − 4.57·29-s + ⋯ |
| L(s) = 1 | + 0.351·2-s − 0.577·3-s − 0.876·4-s − 0.447·5-s − 0.202·6-s − 1.08·7-s − 0.659·8-s + 0.333·9-s − 0.157·10-s − 0.780·11-s + 0.506·12-s − 1.83·13-s − 0.381·14-s + 0.258·15-s + 0.644·16-s + 0.122·17-s + 0.117·18-s + 1.19·19-s + 0.391·20-s + 0.627·21-s − 0.274·22-s + 0.380·24-s + 0.200·25-s − 0.645·26-s − 0.192·27-s + 0.952·28-s − 0.849·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 \) |
| good | 2 | \( 1 - 0.497T + 2T^{2} \) |
| 7 | \( 1 + 2.87T + 7T^{2} \) |
| 11 | \( 1 + 2.59T + 11T^{2} \) |
| 13 | \( 1 + 6.61T + 13T^{2} \) |
| 17 | \( 1 - 0.503T + 17T^{2} \) |
| 19 | \( 1 - 5.19T + 19T^{2} \) |
| 29 | \( 1 + 4.57T + 29T^{2} \) |
| 31 | \( 1 - 5.90T + 31T^{2} \) |
| 37 | \( 1 - 8.53T + 37T^{2} \) |
| 41 | \( 1 - 12.2T + 41T^{2} \) |
| 43 | \( 1 + 1.21T + 43T^{2} \) |
| 47 | \( 1 - 6.34T + 47T^{2} \) |
| 53 | \( 1 - 5.05T + 53T^{2} \) |
| 59 | \( 1 - 4.58T + 59T^{2} \) |
| 61 | \( 1 + 0.573T + 61T^{2} \) |
| 67 | \( 1 + 11.6T + 67T^{2} \) |
| 71 | \( 1 + 7.35T + 71T^{2} \) |
| 73 | \( 1 + 4.52T + 73T^{2} \) |
| 79 | \( 1 + 2.46T + 79T^{2} \) |
| 83 | \( 1 - 1.03T + 83T^{2} \) |
| 89 | \( 1 + 10.4T + 89T^{2} \) |
| 97 | \( 1 - 7.39T + 97T^{2} \) |
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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
−7.55044783449724115488690921797, −6.82262370703392824466466706023, −5.79006645004805912513203410321, −5.49251600890249609105422396602, −4.59917271800536008026183939646, −4.16076137607358761723071728811, −3.06949590561715244598462438428, −2.61787106684608108210060942518, −0.811174498547637629422195065178, 0,
0.811174498547637629422195065178, 2.61787106684608108210060942518, 3.06949590561715244598462438428, 4.16076137607358761723071728811, 4.59917271800536008026183939646, 5.49251600890249609105422396602, 5.79006645004805912513203410321, 6.82262370703392824466466706023, 7.55044783449724115488690921797
