| L(s) = 1 | + 1.63·2-s − 3-s + 0.666·4-s − 3.87·5-s − 1.63·6-s − 7-s − 2.17·8-s + 9-s − 6.32·10-s + 3.71·11-s − 0.666·12-s − 6.24·13-s − 1.63·14-s + 3.87·15-s − 4.88·16-s + 5.99·17-s + 1.63·18-s + 3.05·19-s − 2.58·20-s + 21-s + 6.06·22-s − 1.24·23-s + 2.17·24-s + 9.98·25-s − 10.1·26-s − 27-s − 0.666·28-s + ⋯ |
| L(s) = 1 | + 1.15·2-s − 0.577·3-s + 0.333·4-s − 1.73·5-s − 0.666·6-s − 0.377·7-s − 0.769·8-s + 0.333·9-s − 1.99·10-s + 1.11·11-s − 0.192·12-s − 1.73·13-s − 0.436·14-s + 0.999·15-s − 1.22·16-s + 1.45·17-s + 0.384·18-s + 0.700·19-s − 0.577·20-s + 0.218·21-s + 1.29·22-s − 0.259·23-s + 0.444·24-s + 1.99·25-s − 2.00·26-s − 0.192·27-s − 0.126·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 383 | \( 1 + T \) |
| good | 2 | \( 1 - 1.63T + 2T^{2} \) |
| 5 | \( 1 + 3.87T + 5T^{2} \) |
| 11 | \( 1 - 3.71T + 11T^{2} \) |
| 13 | \( 1 + 6.24T + 13T^{2} \) |
| 17 | \( 1 - 5.99T + 17T^{2} \) |
| 19 | \( 1 - 3.05T + 19T^{2} \) |
| 23 | \( 1 + 1.24T + 23T^{2} \) |
| 29 | \( 1 - 0.671T + 29T^{2} \) |
| 31 | \( 1 - 6.88T + 31T^{2} \) |
| 37 | \( 1 - 0.715T + 37T^{2} \) |
| 41 | \( 1 + 0.510T + 41T^{2} \) |
| 43 | \( 1 - 12.1T + 43T^{2} \) |
| 47 | \( 1 + 6.64T + 47T^{2} \) |
| 53 | \( 1 + 3.69T + 53T^{2} \) |
| 59 | \( 1 - 11.7T + 59T^{2} \) |
| 61 | \( 1 + 6.85T + 61T^{2} \) |
| 67 | \( 1 + 9.35T + 67T^{2} \) |
| 71 | \( 1 - 2.82T + 71T^{2} \) |
| 73 | \( 1 + 0.403T + 73T^{2} \) |
| 79 | \( 1 + 10.4T + 79T^{2} \) |
| 83 | \( 1 + 11.0T + 83T^{2} \) |
| 89 | \( 1 + 11.3T + 89T^{2} \) |
| 97 | \( 1 - 2.94T + 97T^{2} \) |
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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
−7.32603724289496282360339444291, −6.77942299245730732172576998922, −5.93646864181920436466400320767, −5.23280659646197053417920637197, −4.48487638329064351441767713395, −4.13537758122225671190385329173, −3.31852208146630347927296292343, −2.76903264723243772166456368531, −1.01387730726929671010598474401, 0,
1.01387730726929671010598474401, 2.76903264723243772166456368531, 3.31852208146630347927296292343, 4.13537758122225671190385329173, 4.48487638329064351441767713395, 5.23280659646197053417920637197, 5.93646864181920436466400320767, 6.77942299245730732172576998922, 7.32603724289496282360339444291
