| L(s) = 1 | − 1.86·2-s + 1.08·3-s + 1.47·4-s + 0.335·5-s − 2.02·6-s − 1.78·7-s + 0.984·8-s − 1.82·9-s − 0.625·10-s − 2.96·11-s + 1.59·12-s + 5.83·13-s + 3.32·14-s + 0.364·15-s − 4.77·16-s − 17-s + 3.39·18-s + 0.660·19-s + 0.493·20-s − 1.93·21-s + 5.52·22-s + 2.34·23-s + 1.06·24-s − 4.88·25-s − 10.8·26-s − 5.23·27-s − 2.62·28-s + ⋯ |
| L(s) = 1 | − 1.31·2-s + 0.626·3-s + 0.735·4-s + 0.150·5-s − 0.825·6-s − 0.673·7-s + 0.347·8-s − 0.607·9-s − 0.197·10-s − 0.894·11-s + 0.461·12-s + 1.61·13-s + 0.887·14-s + 0.0940·15-s − 1.19·16-s − 0.242·17-s + 0.800·18-s + 0.151·19-s + 0.110·20-s − 0.422·21-s + 1.17·22-s + 0.488·23-s + 0.217·24-s − 0.977·25-s − 2.13·26-s − 1.00·27-s − 0.495·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 17 | \( 1 + T \) |
| 59 | \( 1 + T \) |
| good | 2 | \( 1 + 1.86T + 2T^{2} \) |
| 3 | \( 1 - 1.08T + 3T^{2} \) |
| 5 | \( 1 - 0.335T + 5T^{2} \) |
| 7 | \( 1 + 1.78T + 7T^{2} \) |
| 11 | \( 1 + 2.96T + 11T^{2} \) |
| 13 | \( 1 - 5.83T + 13T^{2} \) |
| 19 | \( 1 - 0.660T + 19T^{2} \) |
| 23 | \( 1 - 2.34T + 23T^{2} \) |
| 29 | \( 1 + 5.05T + 29T^{2} \) |
| 31 | \( 1 + 3.22T + 31T^{2} \) |
| 37 | \( 1 - 7.16T + 37T^{2} \) |
| 41 | \( 1 + 10.1T + 41T^{2} \) |
| 43 | \( 1 + 4.53T + 43T^{2} \) |
| 47 | \( 1 - 3.42T + 47T^{2} \) |
| 53 | \( 1 + 3.90T + 53T^{2} \) |
| 61 | \( 1 - 1.20T + 61T^{2} \) |
| 67 | \( 1 - 7.89T + 67T^{2} \) |
| 71 | \( 1 - 4.37T + 71T^{2} \) |
| 73 | \( 1 + 15.3T + 73T^{2} \) |
| 79 | \( 1 - 0.0651T + 79T^{2} \) |
| 83 | \( 1 + 17.7T + 83T^{2} \) |
| 89 | \( 1 + 10.4T + 89T^{2} \) |
| 97 | \( 1 - 4.30T + 97T^{2} \) |
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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
−9.457257672720964540594716537448, −8.694433025397613210457099925840, −8.208093175829971652166420775258, −7.39741286577896632955863165121, −6.34466176026218670607860545096, −5.42404116543787938341836309332, −3.89316822389454709831134718479, −2.89190215693597742458791996844, −1.67156905981183658174996988806, 0,
1.67156905981183658174996988806, 2.89190215693597742458791996844, 3.89316822389454709831134718479, 5.42404116543787938341836309332, 6.34466176026218670607860545096, 7.39741286577896632955863165121, 8.208093175829971652166420775258, 8.694433025397613210457099925840, 9.457257672720964540594716537448
