L(s) = 1 | − 1.86·2-s − 3-s + 1.47·4-s − 3.19·5-s + 1.86·6-s + 1.30·7-s + 0.978·8-s + 9-s + 5.94·10-s − 0.506·11-s − 1.47·12-s − 3.39·13-s − 2.44·14-s + 3.19·15-s − 4.77·16-s + 17-s − 1.86·18-s + 0.0170·19-s − 4.70·20-s − 1.30·21-s + 0.944·22-s + 3.27·23-s − 0.978·24-s + 5.17·25-s + 6.32·26-s − 27-s + 1.93·28-s + ⋯ |
L(s) = 1 | − 1.31·2-s − 0.577·3-s + 0.737·4-s − 1.42·5-s + 0.761·6-s + 0.494·7-s + 0.345·8-s + 0.333·9-s + 1.88·10-s − 0.152·11-s − 0.425·12-s − 0.941·13-s − 0.652·14-s + 0.823·15-s − 1.19·16-s + 0.242·17-s − 0.439·18-s + 0.00390·19-s − 1.05·20-s − 0.285·21-s + 0.201·22-s + 0.683·23-s − 0.199·24-s + 1.03·25-s + 1.24·26-s − 0.192·27-s + 0.364·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 157 | \( 1 - T \) |
good | 2 | \( 1 + 1.86T + 2T^{2} \) |
| 5 | \( 1 + 3.19T + 5T^{2} \) |
| 7 | \( 1 - 1.30T + 7T^{2} \) |
| 11 | \( 1 + 0.506T + 11T^{2} \) |
| 13 | \( 1 + 3.39T + 13T^{2} \) |
| 19 | \( 1 - 0.0170T + 19T^{2} \) |
| 23 | \( 1 - 3.27T + 23T^{2} \) |
| 29 | \( 1 - 6.50T + 29T^{2} \) |
| 31 | \( 1 + 5.41T + 31T^{2} \) |
| 37 | \( 1 + 1.52T + 37T^{2} \) |
| 41 | \( 1 + 1.68T + 41T^{2} \) |
| 43 | \( 1 + 2.92T + 43T^{2} \) |
| 47 | \( 1 - 6.03T + 47T^{2} \) |
| 53 | \( 1 + 11.2T + 53T^{2} \) |
| 59 | \( 1 + 6.40T + 59T^{2} \) |
| 61 | \( 1 - 6.15T + 61T^{2} \) |
| 67 | \( 1 - 15.0T + 67T^{2} \) |
| 71 | \( 1 + 13.2T + 71T^{2} \) |
| 73 | \( 1 + 0.593T + 73T^{2} \) |
| 79 | \( 1 - 4.20T + 79T^{2} \) |
| 83 | \( 1 - 13.3T + 83T^{2} \) |
| 89 | \( 1 - 8.81T + 89T^{2} \) |
| 97 | \( 1 + 13.1T + 97T^{2} \) |
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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
−7.70439097186161084100500891992, −7.10011419016281202331282338496, −6.48699300557493668013100384098, −5.17114871340417767607759404652, −4.77718393332015482727727803032, −3.99759726913739228883994167108, −3.00676449918127150755294421493, −1.85394988657947730796941188104, −0.827914889115247699465493558470, 0,
0.827914889115247699465493558470, 1.85394988657947730796941188104, 3.00676449918127150755294421493, 3.99759726913739228883994167108, 4.77718393332015482727727803032, 5.17114871340417767607759404652, 6.48699300557493668013100384098, 7.10011419016281202331282338496, 7.70439097186161084100500891992