| L(s) = 1 | − 2-s − 2.61·3-s + 4-s − 3.49·5-s + 2.61·6-s − 0.478·7-s − 8-s + 3.82·9-s + 3.49·10-s − 0.518·11-s − 2.61·12-s + 0.394·13-s + 0.478·14-s + 9.13·15-s + 16-s − 1.63·17-s − 3.82·18-s − 1.57·19-s − 3.49·20-s + 1.25·21-s + 0.518·22-s − 1.44·23-s + 2.61·24-s + 7.21·25-s − 0.394·26-s − 2.16·27-s − 0.478·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.50·3-s + 0.5·4-s − 1.56·5-s + 1.06·6-s − 0.180·7-s − 0.353·8-s + 1.27·9-s + 1.10·10-s − 0.156·11-s − 0.754·12-s + 0.109·13-s + 0.127·14-s + 2.35·15-s + 0.250·16-s − 0.395·17-s − 0.902·18-s − 0.362·19-s − 0.781·20-s + 0.273·21-s + 0.110·22-s − 0.301·23-s + 0.533·24-s + 1.44·25-s − 0.0772·26-s − 0.417·27-s − 0.0904·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8002 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8002 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2183948086\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2183948086\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 4001 | \( 1+O(T) \) |
| good | 3 | \( 1 + 2.61T + 3T^{2} \) |
| 5 | \( 1 + 3.49T + 5T^{2} \) |
| 7 | \( 1 + 0.478T + 7T^{2} \) |
| 11 | \( 1 + 0.518T + 11T^{2} \) |
| 13 | \( 1 - 0.394T + 13T^{2} \) |
| 17 | \( 1 + 1.63T + 17T^{2} \) |
| 19 | \( 1 + 1.57T + 19T^{2} \) |
| 23 | \( 1 + 1.44T + 23T^{2} \) |
| 29 | \( 1 - 6.68T + 29T^{2} \) |
| 31 | \( 1 - 0.0346T + 31T^{2} \) |
| 37 | \( 1 + 0.471T + 37T^{2} \) |
| 41 | \( 1 + 1.81T + 41T^{2} \) |
| 43 | \( 1 - 0.346T + 43T^{2} \) |
| 47 | \( 1 - 13.5T + 47T^{2} \) |
| 53 | \( 1 - 0.660T + 53T^{2} \) |
| 59 | \( 1 - 2.43T + 59T^{2} \) |
| 61 | \( 1 + 6.58T + 61T^{2} \) |
| 67 | \( 1 + 15.0T + 67T^{2} \) |
| 71 | \( 1 + 3.80T + 71T^{2} \) |
| 73 | \( 1 + 4.71T + 73T^{2} \) |
| 79 | \( 1 + 7.31T + 79T^{2} \) |
| 83 | \( 1 + 1.54T + 83T^{2} \) |
| 89 | \( 1 - 15.2T + 89T^{2} \) |
| 97 | \( 1 - 6.19T + 97T^{2} \) |
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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
−7.58858812788797421484276324301, −7.31633529672878446138510537042, −6.40315459240050242412921518010, −6.04279569397923676892242356069, −4.96247497725497426004465300398, −4.43992121043184239293423567402, −3.63318732773040957874564616139, −2.63525091716262029280226937638, −1.23657920386020447922868556930, −0.31577128257494962819048237561,
0.31577128257494962819048237561, 1.23657920386020447922868556930, 2.63525091716262029280226937638, 3.63318732773040957874564616139, 4.43992121043184239293423567402, 4.96247497725497426004465300398, 6.04279569397923676892242356069, 6.40315459240050242412921518010, 7.31633529672878446138510537042, 7.58858812788797421484276324301
