L(s) = 1 | − 1.25·2-s − 2.18·3-s − 0.419·4-s + 4.45·5-s + 2.74·6-s − 4.53·7-s + 3.04·8-s + 1.77·9-s − 5.60·10-s − 0.915·11-s + 0.917·12-s + 3.82·13-s + 5.69·14-s − 9.73·15-s − 2.98·16-s − 17-s − 2.23·18-s + 1.42·19-s − 1.87·20-s + 9.90·21-s + 1.15·22-s − 0.780·23-s − 6.64·24-s + 14.8·25-s − 4.80·26-s + 2.67·27-s + 1.90·28-s + ⋯ |
L(s) = 1 | − 0.888·2-s − 1.26·3-s − 0.209·4-s + 1.99·5-s + 1.12·6-s − 1.71·7-s + 1.07·8-s + 0.591·9-s − 1.77·10-s − 0.276·11-s + 0.264·12-s + 1.06·13-s + 1.52·14-s − 2.51·15-s − 0.746·16-s − 0.242·17-s − 0.525·18-s + 0.327·19-s − 0.418·20-s + 2.16·21-s + 0.245·22-s − 0.162·23-s − 1.35·24-s + 2.97·25-s − 0.942·26-s + 0.515·27-s + 0.359·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5946641401\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5946641401\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 17 | \( 1 + T \) |
| 59 | \( 1 - T \) |
good | 2 | \( 1 + 1.25T + 2T^{2} \) |
| 3 | \( 1 + 2.18T + 3T^{2} \) |
| 5 | \( 1 - 4.45T + 5T^{2} \) |
| 7 | \( 1 + 4.53T + 7T^{2} \) |
| 11 | \( 1 + 0.915T + 11T^{2} \) |
| 13 | \( 1 - 3.82T + 13T^{2} \) |
| 19 | \( 1 - 1.42T + 19T^{2} \) |
| 23 | \( 1 + 0.780T + 23T^{2} \) |
| 29 | \( 1 + 6.11T + 29T^{2} \) |
| 31 | \( 1 + 9.72T + 31T^{2} \) |
| 37 | \( 1 + 7.06T + 37T^{2} \) |
| 41 | \( 1 + 2.81T + 41T^{2} \) |
| 43 | \( 1 - 9.73T + 43T^{2} \) |
| 47 | \( 1 - 7.06T + 47T^{2} \) |
| 53 | \( 1 - 1.70T + 53T^{2} \) |
| 61 | \( 1 - 14.8T + 61T^{2} \) |
| 67 | \( 1 - 2.99T + 67T^{2} \) |
| 71 | \( 1 - 7.46T + 71T^{2} \) |
| 73 | \( 1 + 1.83T + 73T^{2} \) |
| 79 | \( 1 - 7.75T + 79T^{2} \) |
| 83 | \( 1 - 9.33T + 83T^{2} \) |
| 89 | \( 1 - 3.95T + 89T^{2} \) |
| 97 | \( 1 - 1.94T + 97T^{2} \) |
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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
−9.998573986212528646112670703884, −9.214769863436551967753815177838, −8.880940969579053544300017897944, −7.17363853146481787768687165640, −6.46317496411785414008912838623, −5.72463823865093981038831541056, −5.28224623131527969756597005193, −3.64094641572084543362986909079, −2.08427766360933779650479689036, −0.72950024748434165125500007565,
0.72950024748434165125500007565, 2.08427766360933779650479689036, 3.64094641572084543362986909079, 5.28224623131527969756597005193, 5.72463823865093981038831541056, 6.46317496411785414008912838623, 7.17363853146481787768687165640, 8.880940969579053544300017897944, 9.214769863436551967753815177838, 9.998573986212528646112670703884