| L(s) = 1 | − 2-s − 2.79·3-s + 4-s + 2.06·5-s + 2.79·6-s + 3.37·7-s − 8-s + 4.82·9-s − 2.06·10-s + 4.36·11-s − 2.79·12-s − 4.42·13-s − 3.37·14-s − 5.78·15-s + 16-s + 1.92·17-s − 4.82·18-s + 3.03·19-s + 2.06·20-s − 9.44·21-s − 4.36·22-s − 3.72·23-s + 2.79·24-s − 0.727·25-s + 4.42·26-s − 5.10·27-s + 3.37·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.61·3-s + 0.5·4-s + 0.924·5-s + 1.14·6-s + 1.27·7-s − 0.353·8-s + 1.60·9-s − 0.653·10-s + 1.31·11-s − 0.807·12-s − 1.22·13-s − 0.902·14-s − 1.49·15-s + 0.250·16-s + 0.467·17-s − 1.13·18-s + 0.697·19-s + 0.462·20-s − 2.06·21-s − 0.931·22-s − 0.777·23-s + 0.570·24-s − 0.145·25-s + 0.867·26-s − 0.982·27-s + 0.638·28-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
| \(L(1)\) |
\(\approx\) |
\(1.244703299\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.244703299\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.79T + 3T^{2} \) |
| 5 | \( 1 - 2.06T + 5T^{2} \) |
| 7 | \( 1 - 3.37T + 7T^{2} \) |
| 11 | \( 1 - 4.36T + 11T^{2} \) |
| 13 | \( 1 + 4.42T + 13T^{2} \) |
| 17 | \( 1 - 1.92T + 17T^{2} \) |
| 19 | \( 1 - 3.03T + 19T^{2} \) |
| 23 | \( 1 + 3.72T + 23T^{2} \) |
| 29 | \( 1 - 5.04T + 29T^{2} \) |
| 31 | \( 1 - 10.7T + 31T^{2} \) |
| 37 | \( 1 - 10.5T + 37T^{2} \) |
| 41 | \( 1 - 4.85T + 41T^{2} \) |
| 43 | \( 1 + 7.83T + 43T^{2} \) |
| 47 | \( 1 + 6.31T + 47T^{2} \) |
| 53 | \( 1 + 3.65T + 53T^{2} \) |
| 59 | \( 1 - 13.3T + 59T^{2} \) |
| 61 | \( 1 - 3.64T + 61T^{2} \) |
| 67 | \( 1 - 8.76T + 67T^{2} \) |
| 71 | \( 1 - 3.75T + 71T^{2} \) |
| 73 | \( 1 + 5.20T + 73T^{2} \) |
| 79 | \( 1 + 5.14T + 79T^{2} \) |
| 83 | \( 1 + 0.713T + 83T^{2} \) |
| 89 | \( 1 - 6.77T + 89T^{2} \) |
| 97 | \( 1 - 12.5T + 97T^{2} \) |
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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
−8.318909926693070060022434430664, −7.74975114060007293012292468912, −6.73386257413141967773763166247, −6.35262978539597850875959327881, −5.53340872839648852490840655925, −4.92809189926733010704530922851, −4.21082398851628247933336473928, −2.54552971985163227624656121541, −1.51682524805492281265934502895, −0.863676645642871025235367108005,
0.863676645642871025235367108005, 1.51682524805492281265934502895, 2.54552971985163227624656121541, 4.21082398851628247933336473928, 4.92809189926733010704530922851, 5.53340872839648852490840655925, 6.35262978539597850875959327881, 6.73386257413141967773763166247, 7.74975114060007293012292468912, 8.318909926693070060022434430664
