| L(s) = 1 | − 2.76·2-s − 0.782·3-s + 5.63·4-s + 4.06·5-s + 2.16·6-s + 1.66·7-s − 10.0·8-s − 2.38·9-s − 11.2·10-s + 5.36·11-s − 4.41·12-s + 4.72·13-s − 4.59·14-s − 3.17·15-s + 16.5·16-s − 0.338·17-s + 6.59·18-s − 2.39·19-s + 22.8·20-s − 1.30·21-s − 14.8·22-s + 1.52·23-s + 7.86·24-s + 11.4·25-s − 13.0·26-s + 4.21·27-s + 9.37·28-s + ⋯ |
| L(s) = 1 | − 1.95·2-s − 0.451·3-s + 2.81·4-s + 1.81·5-s + 0.883·6-s + 0.628·7-s − 3.55·8-s − 0.795·9-s − 3.54·10-s + 1.61·11-s − 1.27·12-s + 1.31·13-s − 1.22·14-s − 0.820·15-s + 4.12·16-s − 0.0821·17-s + 1.55·18-s − 0.548·19-s + 5.11·20-s − 0.284·21-s − 3.16·22-s + 0.317·23-s + 1.60·24-s + 2.29·25-s − 2.56·26-s + 0.811·27-s + 1.77·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.243351577\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.243351577\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 4001 | \( 1+O(T) \) |
| good | 2 | \( 1 + 2.76T + 2T^{2} \) |
| 3 | \( 1 + 0.782T + 3T^{2} \) |
| 5 | \( 1 - 4.06T + 5T^{2} \) |
| 7 | \( 1 - 1.66T + 7T^{2} \) |
| 11 | \( 1 - 5.36T + 11T^{2} \) |
| 13 | \( 1 - 4.72T + 13T^{2} \) |
| 17 | \( 1 + 0.338T + 17T^{2} \) |
| 19 | \( 1 + 2.39T + 19T^{2} \) |
| 23 | \( 1 - 1.52T + 23T^{2} \) |
| 29 | \( 1 + 2.87T + 29T^{2} \) |
| 31 | \( 1 - 7.10T + 31T^{2} \) |
| 37 | \( 1 - 7.03T + 37T^{2} \) |
| 41 | \( 1 + 7.06T + 41T^{2} \) |
| 43 | \( 1 + 10.7T + 43T^{2} \) |
| 47 | \( 1 + 7.82T + 47T^{2} \) |
| 53 | \( 1 - 13.0T + 53T^{2} \) |
| 59 | \( 1 - 0.352T + 59T^{2} \) |
| 61 | \( 1 + 2.23T + 61T^{2} \) |
| 67 | \( 1 - 8.18T + 67T^{2} \) |
| 71 | \( 1 - 9.16T + 71T^{2} \) |
| 73 | \( 1 + 9.95T + 73T^{2} \) |
| 79 | \( 1 - 14.1T + 79T^{2} \) |
| 83 | \( 1 + 2.57T + 83T^{2} \) |
| 89 | \( 1 - 15.4T + 89T^{2} \) |
| 97 | \( 1 - 2.79T + 97T^{2} \) |
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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
−8.611451523562219188660092945902, −8.162256662934724363611508458382, −6.74157705976395324965641711153, −6.44249763571049670715921406664, −6.01558017309345214091914271609, −5.09856258016982993562850627552, −3.42622350682267399124785557953, −2.33441621603393359648992819788, −1.56961674514797472934814529366, −0.975566362556027713750780891742,
0.975566362556027713750780891742, 1.56961674514797472934814529366, 2.33441621603393359648992819788, 3.42622350682267399124785557953, 5.09856258016982993562850627552, 6.01558017309345214091914271609, 6.44249763571049670715921406664, 6.74157705976395324965641711153, 8.162256662934724363611508458382, 8.611451523562219188660092945902
