L(s) = 1 | − 2-s − 3.06·3-s + 4-s + 4.19·5-s + 3.06·6-s + 2.77·7-s − 8-s + 6.39·9-s − 4.19·10-s − 1.46·11-s − 3.06·12-s + 5.16·13-s − 2.77·14-s − 12.8·15-s + 16-s − 2.76·17-s − 6.39·18-s − 5.73·19-s + 4.19·20-s − 8.50·21-s + 1.46·22-s + 0.445·23-s + 3.06·24-s + 12.5·25-s − 5.16·26-s − 10.4·27-s + 2.77·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.76·3-s + 0.5·4-s + 1.87·5-s + 1.25·6-s + 1.04·7-s − 0.353·8-s + 2.13·9-s − 1.32·10-s − 0.440·11-s − 0.884·12-s + 1.43·13-s − 0.741·14-s − 3.31·15-s + 0.250·16-s − 0.670·17-s − 1.50·18-s − 1.31·19-s + 0.937·20-s − 1.85·21-s + 0.311·22-s + 0.0929·23-s + 0.625·24-s + 2.51·25-s − 1.01·26-s − 2.00·27-s + 0.524·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.311140129\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.311140129\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 2003 | \( 1 - T \) |
good | 3 | \( 1 + 3.06T + 3T^{2} \) |
| 5 | \( 1 - 4.19T + 5T^{2} \) |
| 7 | \( 1 - 2.77T + 7T^{2} \) |
| 11 | \( 1 + 1.46T + 11T^{2} \) |
| 13 | \( 1 - 5.16T + 13T^{2} \) |
| 17 | \( 1 + 2.76T + 17T^{2} \) |
| 19 | \( 1 + 5.73T + 19T^{2} \) |
| 23 | \( 1 - 0.445T + 23T^{2} \) |
| 29 | \( 1 - 2.61T + 29T^{2} \) |
| 31 | \( 1 + 4.88T + 31T^{2} \) |
| 37 | \( 1 - 6.83T + 37T^{2} \) |
| 41 | \( 1 - 4.60T + 41T^{2} \) |
| 43 | \( 1 - 4.71T + 43T^{2} \) |
| 47 | \( 1 - 1.83T + 47T^{2} \) |
| 53 | \( 1 + 0.469T + 53T^{2} \) |
| 59 | \( 1 - 11.6T + 59T^{2} \) |
| 61 | \( 1 + 4.43T + 61T^{2} \) |
| 67 | \( 1 + 4.75T + 67T^{2} \) |
| 71 | \( 1 - 12.6T + 71T^{2} \) |
| 73 | \( 1 + 6.74T + 73T^{2} \) |
| 79 | \( 1 + 0.365T + 79T^{2} \) |
| 83 | \( 1 - 0.362T + 83T^{2} \) |
| 89 | \( 1 - 8.83T + 89T^{2} \) |
| 97 | \( 1 + 2.40T + 97T^{2} \) |
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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
−8.652661248596757354697604550849, −7.64428846440047935778639389623, −6.58893636214202392740108717836, −6.25531526883398688543233997184, −5.67441786528835391326357527159, −5.03095616858667649517368090717, −4.20224167837998026700280722216, −2.37714777804665680878810432601, −1.65028201945827232728238736662, −0.860304107599772441966975014112,
0.860304107599772441966975014112, 1.65028201945827232728238736662, 2.37714777804665680878810432601, 4.20224167837998026700280722216, 5.03095616858667649517368090717, 5.67441786528835391326357527159, 6.25531526883398688543233997184, 6.58893636214202392740108717836, 7.64428846440047935778639389623, 8.652661248596757354697604550849