| L(s) = 1 | − 2.40·2-s + 0.531·3-s + 3.77·4-s + 3.68·5-s − 1.27·6-s − 3.72·7-s − 4.25·8-s − 2.71·9-s − 8.85·10-s − 4.55·11-s + 2.00·12-s − 0.961·13-s + 8.94·14-s + 1.95·15-s + 2.68·16-s − 4.53·17-s + 6.52·18-s − 1.62·19-s + 13.8·20-s − 1.97·21-s + 10.9·22-s + 0.930·23-s − 2.26·24-s + 8.57·25-s + 2.31·26-s − 3.03·27-s − 14.0·28-s + ⋯ |
| L(s) = 1 | − 1.69·2-s + 0.306·3-s + 1.88·4-s + 1.64·5-s − 0.521·6-s − 1.40·7-s − 1.50·8-s − 0.905·9-s − 2.79·10-s − 1.37·11-s + 0.578·12-s − 0.266·13-s + 2.39·14-s + 0.505·15-s + 0.670·16-s − 1.09·17-s + 1.53·18-s − 0.373·19-s + 3.10·20-s − 0.431·21-s + 2.33·22-s + 0.193·23-s − 0.461·24-s + 1.71·25-s + 0.453·26-s − 0.584·27-s − 2.65·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5560833569\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5560833569\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 4003 | \( 1+O(T) \) |
| good | 2 | \( 1 + 2.40T + 2T^{2} \) |
| 3 | \( 1 - 0.531T + 3T^{2} \) |
| 5 | \( 1 - 3.68T + 5T^{2} \) |
| 7 | \( 1 + 3.72T + 7T^{2} \) |
| 11 | \( 1 + 4.55T + 11T^{2} \) |
| 13 | \( 1 + 0.961T + 13T^{2} \) |
| 17 | \( 1 + 4.53T + 17T^{2} \) |
| 19 | \( 1 + 1.62T + 19T^{2} \) |
| 23 | \( 1 - 0.930T + 23T^{2} \) |
| 29 | \( 1 - 6.79T + 29T^{2} \) |
| 31 | \( 1 + 5.57T + 31T^{2} \) |
| 37 | \( 1 - 2.73T + 37T^{2} \) |
| 41 | \( 1 + 9.14T + 41T^{2} \) |
| 43 | \( 1 - 1.82T + 43T^{2} \) |
| 47 | \( 1 - 0.00696T + 47T^{2} \) |
| 53 | \( 1 + 8.51T + 53T^{2} \) |
| 59 | \( 1 - 5.88T + 59T^{2} \) |
| 61 | \( 1 - 13.4T + 61T^{2} \) |
| 67 | \( 1 - 12.6T + 67T^{2} \) |
| 71 | \( 1 - 13.4T + 71T^{2} \) |
| 73 | \( 1 - 5.62T + 73T^{2} \) |
| 79 | \( 1 + 8.28T + 79T^{2} \) |
| 83 | \( 1 + 7.24T + 83T^{2} \) |
| 89 | \( 1 + 4.63T + 89T^{2} \) |
| 97 | \( 1 - 16.1T + 97T^{2} \) |
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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
−8.665419306851321559385222591937, −8.043331978784136630372263058409, −6.91848294530372147333699720761, −6.54636033541698097176575123077, −5.80328781080756381414345903470, −5.00668303457922417068683199841, −3.23620657370204347672929822590, −2.43761121132861130978021814192, −2.11557391575798292935035548808, −0.50050432603200724840470874296,
0.50050432603200724840470874296, 2.11557391575798292935035548808, 2.43761121132861130978021814192, 3.23620657370204347672929822590, 5.00668303457922417068683199841, 5.80328781080756381414345903470, 6.54636033541698097176575123077, 6.91848294530372147333699720761, 8.043331978784136630372263058409, 8.665419306851321559385222591937
