| L(s) = 1 | − 2-s + 3.32·3-s − 4-s − 2.39·5-s − 3.32·6-s − 3.32·7-s + 3·8-s + 8.04·9-s + 2.39·10-s − 3.32·12-s − 2.92·13-s + 3.32·14-s − 7.97·15-s − 16-s − 17-s − 8.04·18-s − 5.32·19-s + 2.39·20-s − 11.0·21-s − 0.925·23-s + 9.97·24-s + 0.751·25-s + 2.92·26-s + 16.7·27-s + 3.32·28-s + 4.24·29-s + 7.97·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.91·3-s − 0.5·4-s − 1.07·5-s − 1.35·6-s − 1.25·7-s + 1.06·8-s + 2.68·9-s + 0.758·10-s − 0.959·12-s − 0.811·13-s + 0.888·14-s − 2.05·15-s − 0.250·16-s − 0.242·17-s − 1.89·18-s − 1.22·19-s + 0.536·20-s − 2.41·21-s − 0.192·23-s + 2.03·24-s + 0.150·25-s + 0.573·26-s + 3.22·27-s + 0.628·28-s + 0.788·29-s + 1.45·30-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 | \( 1 + T + 2T^{2} \) |
| 3 | \( 1 - 3.32T + 3T^{2} \) |
| 5 | \( 1 + 2.39T + 5T^{2} \) |
| 7 | \( 1 + 3.32T + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 2.92T + 13T^{2} \) |
| 19 | \( 1 + 5.32T + 19T^{2} \) |
| 23 | \( 1 + 0.925T + 23T^{2} \) |
| 29 | \( 1 - 4.24T + 29T^{2} \) |
| 31 | \( 1 + 9.72T + 31T^{2} \) |
| 37 | \( 1 + 1.85T + 37T^{2} \) |
| 41 | \( 1 + 7.69T + 41T^{2} \) |
| 43 | \( 1 - 1.72T + 43T^{2} \) |
| 47 | \( 1 + 6.51T + 47T^{2} \) |
| 53 | \( 1 + 10.3T + 53T^{2} \) |
| 61 | \( 1 - 8.36T + 61T^{2} \) |
| 67 | \( 1 - 9.29T + 67T^{2} \) |
| 71 | \( 1 + 12.6T + 71T^{2} \) |
| 73 | \( 1 + 6.64T + 73T^{2} \) |
| 79 | \( 1 - 8.91T + 79T^{2} \) |
| 83 | \( 1 - 7.44T + 83T^{2} \) |
| 89 | \( 1 + 5.07T + 89T^{2} \) |
| 97 | \( 1 - 0.427T + 97T^{2} \) |
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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
−9.428506717493217731392205007447, −8.678047095075695089710895760373, −8.147329665285259247422953148926, −7.40034123956230788247987078711, −6.72334145877086288464852472797, −4.70476343882007130395904287321, −3.88028375575387295947059759194, −3.23473304251372872166886501377, −1.98381012843150294068347337598, 0,
1.98381012843150294068347337598, 3.23473304251372872166886501377, 3.88028375575387295947059759194, 4.70476343882007130395904287321, 6.72334145877086288464852472797, 7.40034123956230788247987078711, 8.147329665285259247422953148926, 8.678047095075695089710895760373, 9.428506717493217731392205007447
