| L(s) = 1 | + 2-s − 0.381·3-s + 4-s − 3.61·5-s − 0.381·6-s − 3·7-s + 8-s − 2.85·9-s − 3.61·10-s − 4.61·11-s − 0.381·12-s − 3·13-s − 3·14-s + 1.38·15-s + 16-s + 1.47·17-s − 2.85·18-s − 3·19-s − 3.61·20-s + 1.14·21-s − 4.61·22-s − 7.47·23-s − 0.381·24-s + 8.09·25-s − 3·26-s + 2.23·27-s − 3·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.220·3-s + 0.5·4-s − 1.61·5-s − 0.155·6-s − 1.13·7-s + 0.353·8-s − 0.951·9-s − 1.14·10-s − 1.39·11-s − 0.110·12-s − 0.832·13-s − 0.801·14-s + 0.356·15-s + 0.250·16-s + 0.357·17-s − 0.672·18-s − 0.688·19-s − 0.809·20-s + 0.250·21-s − 0.984·22-s − 1.55·23-s − 0.0779·24-s + 1.61·25-s − 0.588·26-s + 0.430·27-s − 0.566·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6038 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6038 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3019 | \( 1+O(T) \) |
| good | 3 | \( 1 + 0.381T + 3T^{2} \) |
| 5 | \( 1 + 3.61T + 5T^{2} \) |
| 7 | \( 1 + 3T + 7T^{2} \) |
| 11 | \( 1 + 4.61T + 11T^{2} \) |
| 13 | \( 1 + 3T + 13T^{2} \) |
| 17 | \( 1 - 1.47T + 17T^{2} \) |
| 19 | \( 1 + 3T + 19T^{2} \) |
| 23 | \( 1 + 7.47T + 23T^{2} \) |
| 29 | \( 1 + 4.76T + 29T^{2} \) |
| 31 | \( 1 - 5.09T + 31T^{2} \) |
| 37 | \( 1 - 4.85T + 37T^{2} \) |
| 41 | \( 1 + 11.1T + 41T^{2} \) |
| 43 | \( 1 + 4.76T + 43T^{2} \) |
| 47 | \( 1 + 8.56T + 47T^{2} \) |
| 53 | \( 1 + 9T + 53T^{2} \) |
| 59 | \( 1 + 14.2T + 59T^{2} \) |
| 61 | \( 1 + 4.61T + 61T^{2} \) |
| 67 | \( 1 + 4.85T + 67T^{2} \) |
| 71 | \( 1 - 5.94T + 71T^{2} \) |
| 73 | \( 1 + 12.7T + 73T^{2} \) |
| 79 | \( 1 - 4.85T + 79T^{2} \) |
| 83 | \( 1 - 15.9T + 83T^{2} \) |
| 89 | \( 1 - 7.79T + 89T^{2} \) |
| 97 | \( 1 - 18.2T + 97T^{2} \) |
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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
−7.49637706986989643634034703940, −6.46178064959430146484122118225, −6.04002451607903648633524029033, −4.99050022039682986454604642705, −4.56388580421297763359868069522, −3.37692936114172083921266340471, −3.22422217533988060994973477349, −2.20540295289471703522520176571, 0, 0,
2.20540295289471703522520176571, 3.22422217533988060994973477349, 3.37692936114172083921266340471, 4.56388580421297763359868069522, 4.99050022039682986454604642705, 6.04002451607903648633524029033, 6.46178064959430146484122118225, 7.49637706986989643634034703940
