| L(s) = 1 | + 0.762·2-s + 3-s − 1.41·4-s − 4.10·5-s + 0.762·6-s − 2.37·7-s − 2.60·8-s + 9-s − 3.12·10-s − 4.76·11-s − 1.41·12-s + 5.19·13-s − 1.81·14-s − 4.10·15-s + 0.848·16-s − 2.80·17-s + 0.762·18-s − 0.644·19-s + 5.82·20-s − 2.37·21-s − 3.63·22-s + 23-s − 2.60·24-s + 11.8·25-s + 3.95·26-s + 27-s + 3.36·28-s + ⋯ |
| L(s) = 1 | + 0.539·2-s + 0.577·3-s − 0.709·4-s − 1.83·5-s + 0.311·6-s − 0.898·7-s − 0.921·8-s + 0.333·9-s − 0.989·10-s − 1.43·11-s − 0.409·12-s + 1.43·13-s − 0.484·14-s − 1.05·15-s + 0.212·16-s − 0.681·17-s + 0.179·18-s − 0.147·19-s + 1.30·20-s − 0.518·21-s − 0.775·22-s + 0.208·23-s − 0.532·24-s + 2.36·25-s + 0.776·26-s + 0.192·27-s + 0.636·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9207495988\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9207495988\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 - 0.762T + 2T^{2} \) |
| 5 | \( 1 + 4.10T + 5T^{2} \) |
| 7 | \( 1 + 2.37T + 7T^{2} \) |
| 11 | \( 1 + 4.76T + 11T^{2} \) |
| 13 | \( 1 - 5.19T + 13T^{2} \) |
| 17 | \( 1 + 2.80T + 17T^{2} \) |
| 19 | \( 1 + 0.644T + 19T^{2} \) |
| 31 | \( 1 - 0.195T + 31T^{2} \) |
| 37 | \( 1 - 4.46T + 37T^{2} \) |
| 41 | \( 1 - 2.20T + 41T^{2} \) |
| 43 | \( 1 - 10.7T + 43T^{2} \) |
| 47 | \( 1 + 10.3T + 47T^{2} \) |
| 53 | \( 1 + 3.71T + 53T^{2} \) |
| 59 | \( 1 + 6.79T + 59T^{2} \) |
| 61 | \( 1 - 12.5T + 61T^{2} \) |
| 67 | \( 1 + 2.80T + 67T^{2} \) |
| 71 | \( 1 - 14.2T + 71T^{2} \) |
| 73 | \( 1 + 7.86T + 73T^{2} \) |
| 79 | \( 1 + 9.62T + 79T^{2} \) |
| 83 | \( 1 - 14.6T + 83T^{2} \) |
| 89 | \( 1 + 8.72T + 89T^{2} \) |
| 97 | \( 1 + 13.7T + 97T^{2} \) |
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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
−8.931481352873067749724104855258, −8.295012199202086507446359417190, −7.85792552587958297889545522361, −6.85396862356667397195311471844, −5.93500903893389320141953815665, −4.78905813926348571240818210353, −4.12166420058441962583355174135, −3.42164501327251304104940673790, −2.82267828433825748124510266546, −0.55379191183042260852570291370,
0.55379191183042260852570291370, 2.82267828433825748124510266546, 3.42164501327251304104940673790, 4.12166420058441962583355174135, 4.78905813926348571240818210353, 5.93500903893389320141953815665, 6.85396862356667397195311471844, 7.85792552587958297889545522361, 8.295012199202086507446359417190, 8.931481352873067749724104855258
