| L(s) = 1 | − 2-s + 0.961·3-s + 4-s + 2.47·5-s − 0.961·6-s − 3.64·7-s − 8-s − 2.07·9-s − 2.47·10-s − 0.534·11-s + 0.961·12-s + 6.07·13-s + 3.64·14-s + 2.38·15-s + 16-s − 5.92·17-s + 2.07·18-s + 0.706·19-s + 2.47·20-s − 3.50·21-s + 0.534·22-s + 2.43·23-s − 0.961·24-s + 1.14·25-s − 6.07·26-s − 4.87·27-s − 3.64·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.554·3-s + 0.5·4-s + 1.10·5-s − 0.392·6-s − 1.37·7-s − 0.353·8-s − 0.692·9-s − 0.784·10-s − 0.161·11-s + 0.277·12-s + 1.68·13-s + 0.974·14-s + 0.615·15-s + 0.250·16-s − 1.43·17-s + 0.489·18-s + 0.162·19-s + 0.554·20-s − 0.764·21-s + 0.114·22-s + 0.508·23-s − 0.196·24-s + 0.229·25-s − 1.19·26-s − 0.938·27-s − 0.689·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 2003 | \( 1 + T \) |
| good | 3 | \( 1 - 0.961T + 3T^{2} \) |
| 5 | \( 1 - 2.47T + 5T^{2} \) |
| 7 | \( 1 + 3.64T + 7T^{2} \) |
| 11 | \( 1 + 0.534T + 11T^{2} \) |
| 13 | \( 1 - 6.07T + 13T^{2} \) |
| 17 | \( 1 + 5.92T + 17T^{2} \) |
| 19 | \( 1 - 0.706T + 19T^{2} \) |
| 23 | \( 1 - 2.43T + 23T^{2} \) |
| 29 | \( 1 + 5.02T + 29T^{2} \) |
| 31 | \( 1 - 4.49T + 31T^{2} \) |
| 37 | \( 1 + 5.93T + 37T^{2} \) |
| 41 | \( 1 + 7.28T + 41T^{2} \) |
| 43 | \( 1 - 7.34T + 43T^{2} \) |
| 47 | \( 1 - 1.28T + 47T^{2} \) |
| 53 | \( 1 - 7.69T + 53T^{2} \) |
| 59 | \( 1 + 8.59T + 59T^{2} \) |
| 61 | \( 1 + 3.03T + 61T^{2} \) |
| 67 | \( 1 + 6.51T + 67T^{2} \) |
| 71 | \( 1 + 8.18T + 71T^{2} \) |
| 73 | \( 1 + 1.72T + 73T^{2} \) |
| 79 | \( 1 - 5.13T + 79T^{2} \) |
| 83 | \( 1 + 10.8T + 83T^{2} \) |
| 89 | \( 1 + 2.56T + 89T^{2} \) |
| 97 | \( 1 + 3.12T + 97T^{2} \) |
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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
−8.433813009441560889690934767827, −7.35594701742413643663991268193, −6.47820390460229721539707698554, −6.13673195271272800295263900674, −5.42952110971139560296502401910, −3.96175230335543746654986849276, −3.13630285815638072503638301180, −2.46815846242975550754444404778, −1.49368603366334255675491277744, 0,
1.49368603366334255675491277744, 2.46815846242975550754444404778, 3.13630285815638072503638301180, 3.96175230335543746654986849276, 5.42952110971139560296502401910, 6.13673195271272800295263900674, 6.47820390460229721539707698554, 7.35594701742413643663991268193, 8.433813009441560889690934767827
