| L(s) = 1 | − 2-s − 0.246·3-s + 4-s − 3.43·5-s + 0.246·6-s + 1.57·7-s − 8-s − 2.93·9-s + 3.43·10-s − 4.37·11-s − 0.246·12-s − 4.54·13-s − 1.57·14-s + 0.847·15-s + 16-s + 2.21·17-s + 2.93·18-s − 0.423·19-s − 3.43·20-s − 0.387·21-s + 4.37·22-s − 6.15·23-s + 0.246·24-s + 6.77·25-s + 4.54·26-s + 1.46·27-s + 1.57·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.142·3-s + 0.5·4-s − 1.53·5-s + 0.100·6-s + 0.593·7-s − 0.353·8-s − 0.979·9-s + 1.08·10-s − 1.31·11-s − 0.0712·12-s − 1.26·13-s − 0.419·14-s + 0.218·15-s + 0.250·16-s + 0.537·17-s + 0.692·18-s − 0.0971·19-s − 0.767·20-s − 0.0846·21-s + 0.932·22-s − 1.28·23-s + 0.0504·24-s + 1.35·25-s + 0.891·26-s + 0.282·27-s + 0.296·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1682 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1682 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3941037310\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3941037310\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 29 | \( 1 \) |
| good | 3 | \( 1 + 0.246T + 3T^{2} \) |
| 5 | \( 1 + 3.43T + 5T^{2} \) |
| 7 | \( 1 - 1.57T + 7T^{2} \) |
| 11 | \( 1 + 4.37T + 11T^{2} \) |
| 13 | \( 1 + 4.54T + 13T^{2} \) |
| 17 | \( 1 - 2.21T + 17T^{2} \) |
| 19 | \( 1 + 0.423T + 19T^{2} \) |
| 23 | \( 1 + 6.15T + 23T^{2} \) |
| 31 | \( 1 - 7.46T + 31T^{2} \) |
| 37 | \( 1 + 1.58T + 37T^{2} \) |
| 41 | \( 1 - 4.56T + 41T^{2} \) |
| 43 | \( 1 - 9.26T + 43T^{2} \) |
| 47 | \( 1 + 4.31T + 47T^{2} \) |
| 53 | \( 1 - 6.53T + 53T^{2} \) |
| 59 | \( 1 + 14.5T + 59T^{2} \) |
| 61 | \( 1 + 6.16T + 61T^{2} \) |
| 67 | \( 1 + 8.23T + 67T^{2} \) |
| 71 | \( 1 - 12.6T + 71T^{2} \) |
| 73 | \( 1 - 8.64T + 73T^{2} \) |
| 79 | \( 1 - 6.44T + 79T^{2} \) |
| 83 | \( 1 - 0.615T + 83T^{2} \) |
| 89 | \( 1 + 5.06T + 89T^{2} \) |
| 97 | \( 1 - 8.13T + 97T^{2} \) |
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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
−9.257763040917182905646143915375, −8.196319582519544556849516592761, −7.906361343104150912745068288289, −7.45722004484830576717904842908, −6.21109743904262406600214473936, −5.19529242873265559168046078591, −4.43352077501244876439543065545, −3.17740087017611994146166178589, −2.34252973297914087597079201061, −0.45563952397857100426204094077,
0.45563952397857100426204094077, 2.34252973297914087597079201061, 3.17740087017611994146166178589, 4.43352077501244876439543065545, 5.19529242873265559168046078591, 6.21109743904262406600214473936, 7.45722004484830576717904842908, 7.906361343104150912745068288289, 8.196319582519544556849516592761, 9.257763040917182905646143915375
