L(s) = 1 | + 0.403·2-s + 3-s − 1.83·4-s − 5-s + 0.403·6-s + 0.330·7-s − 1.54·8-s + 9-s − 0.403·10-s + 0.933·11-s − 1.83·12-s − 4.15·13-s + 0.133·14-s − 15-s + 3.04·16-s − 0.316·17-s + 0.403·18-s + 2.03·19-s + 1.83·20-s + 0.330·21-s + 0.377·22-s + 1.34·23-s − 1.54·24-s + 25-s − 1.67·26-s + 27-s − 0.607·28-s + ⋯ |
L(s) = 1 | + 0.285·2-s + 0.577·3-s − 0.918·4-s − 0.447·5-s + 0.164·6-s + 0.124·7-s − 0.547·8-s + 0.333·9-s − 0.127·10-s + 0.281·11-s − 0.530·12-s − 1.15·13-s + 0.0356·14-s − 0.258·15-s + 0.761·16-s − 0.0768·17-s + 0.0952·18-s + 0.466·19-s + 0.410·20-s + 0.0721·21-s + 0.0803·22-s + 0.281·23-s − 0.316·24-s + 0.200·25-s − 0.329·26-s + 0.192·27-s − 0.114·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 401 | \( 1 - T \) |
good | 2 | \( 1 - 0.403T + 2T^{2} \) |
| 7 | \( 1 - 0.330T + 7T^{2} \) |
| 11 | \( 1 - 0.933T + 11T^{2} \) |
| 13 | \( 1 + 4.15T + 13T^{2} \) |
| 17 | \( 1 + 0.316T + 17T^{2} \) |
| 19 | \( 1 - 2.03T + 19T^{2} \) |
| 23 | \( 1 - 1.34T + 23T^{2} \) |
| 29 | \( 1 - 0.499T + 29T^{2} \) |
| 31 | \( 1 - 2.54T + 31T^{2} \) |
| 37 | \( 1 + 6.91T + 37T^{2} \) |
| 41 | \( 1 - 6.76T + 41T^{2} \) |
| 43 | \( 1 - 10.2T + 43T^{2} \) |
| 47 | \( 1 + 12.3T + 47T^{2} \) |
| 53 | \( 1 - 0.790T + 53T^{2} \) |
| 59 | \( 1 + 8.46T + 59T^{2} \) |
| 61 | \( 1 - 3.59T + 61T^{2} \) |
| 67 | \( 1 + 5.69T + 67T^{2} \) |
| 71 | \( 1 - 10.3T + 71T^{2} \) |
| 73 | \( 1 + 8.44T + 73T^{2} \) |
| 79 | \( 1 - 7.10T + 79T^{2} \) |
| 83 | \( 1 + 8.85T + 83T^{2} \) |
| 89 | \( 1 - 2.47T + 89T^{2} \) |
| 97 | \( 1 + 8.94T + 97T^{2} \) |
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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
−7.82168365693021670414109571838, −7.17776444276278544734586409081, −6.31274492943501720660573990341, −5.31260740484131249293977572008, −4.73836461054798905439521732166, −4.09820171153821248929258697875, −3.30002488714173353034651158970, −2.57700579476286866360917838007, −1.27890132655505043110066760710, 0,
1.27890132655505043110066760710, 2.57700579476286866360917838007, 3.30002488714173353034651158970, 4.09820171153821248929258697875, 4.73836461054798905439521732166, 5.31260740484131249293977572008, 6.31274492943501720660573990341, 7.17776444276278544734586409081, 7.82168365693021670414109571838