| L(s) = 1 | − 3-s + 3.54·5-s + 2.31·7-s + 9-s − 0.0711·11-s − 0.620·13-s − 3.54·15-s − 4.97·17-s − 6.83·19-s − 2.31·21-s + 23-s + 7.53·25-s − 27-s − 29-s − 2.38·31-s + 0.0711·33-s + 8.18·35-s − 7.78·37-s + 0.620·39-s − 6.42·41-s + 1.04·43-s + 3.54·45-s + 5.51·47-s − 1.66·49-s + 4.97·51-s − 5.37·53-s − 0.252·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.58·5-s + 0.873·7-s + 0.333·9-s − 0.0214·11-s − 0.172·13-s − 0.914·15-s − 1.20·17-s − 1.56·19-s − 0.504·21-s + 0.208·23-s + 1.50·25-s − 0.192·27-s − 0.185·29-s − 0.428·31-s + 0.0123·33-s + 1.38·35-s − 1.28·37-s + 0.0994·39-s − 1.00·41-s + 0.159·43-s + 0.527·45-s + 0.804·47-s − 0.237·49-s + 0.697·51-s − 0.738·53-s − 0.0339·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 5 | \( 1 - 3.54T + 5T^{2} \) |
| 7 | \( 1 - 2.31T + 7T^{2} \) |
| 11 | \( 1 + 0.0711T + 11T^{2} \) |
| 13 | \( 1 + 0.620T + 13T^{2} \) |
| 17 | \( 1 + 4.97T + 17T^{2} \) |
| 19 | \( 1 + 6.83T + 19T^{2} \) |
| 31 | \( 1 + 2.38T + 31T^{2} \) |
| 37 | \( 1 + 7.78T + 37T^{2} \) |
| 41 | \( 1 + 6.42T + 41T^{2} \) |
| 43 | \( 1 - 1.04T + 43T^{2} \) |
| 47 | \( 1 - 5.51T + 47T^{2} \) |
| 53 | \( 1 + 5.37T + 53T^{2} \) |
| 59 | \( 1 - 0.604T + 59T^{2} \) |
| 61 | \( 1 + 9.02T + 61T^{2} \) |
| 67 | \( 1 + 12.1T + 67T^{2} \) |
| 71 | \( 1 + 3.52T + 71T^{2} \) |
| 73 | \( 1 + 4.78T + 73T^{2} \) |
| 79 | \( 1 - 1.72T + 79T^{2} \) |
| 83 | \( 1 + 12.0T + 83T^{2} \) |
| 89 | \( 1 + 8.77T + 89T^{2} \) |
| 97 | \( 1 - 5.44T + 97T^{2} \) |
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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
−7.26472810657483514097727577465, −6.67263482392236936748134000349, −6.06383475962409763150035617355, −5.46084044284973293403172402658, −4.76620806827764248985572271360, −4.22316973867577100294483297070, −2.87069718912065476508359638693, −1.90451609430070012392623032903, −1.61551138226079428243645151775, 0,
1.61551138226079428243645151775, 1.90451609430070012392623032903, 2.87069718912065476508359638693, 4.22316973867577100294483297070, 4.76620806827764248985572271360, 5.46084044284973293403172402658, 6.06383475962409763150035617355, 6.67263482392236936748134000349, 7.26472810657483514097727577465
