| L(s) = 1 | − 0.387·3-s − 1.46·5-s − 0.194·7-s − 2.84·9-s − 1.87·11-s + 6.15·13-s + 0.569·15-s − 5.37·17-s + 3.78·19-s + 0.0755·21-s + 3.43·23-s − 2.84·25-s + 2.26·27-s + 0.0833·29-s + 3.95·31-s + 0.726·33-s + 0.286·35-s + 3.24·37-s − 2.38·39-s − 12.5·41-s − 2.32·43-s + 4.18·45-s + 8.88·47-s − 6.96·49-s + 2.08·51-s + 7.04·53-s + 2.75·55-s + ⋯ |
| L(s) = 1 | − 0.223·3-s − 0.657·5-s − 0.0735·7-s − 0.949·9-s − 0.564·11-s + 1.70·13-s + 0.147·15-s − 1.30·17-s + 0.867·19-s + 0.0164·21-s + 0.715·23-s − 0.568·25-s + 0.436·27-s + 0.0154·29-s + 0.709·31-s + 0.126·33-s + 0.0483·35-s + 0.533·37-s − 0.382·39-s − 1.95·41-s − 0.354·43-s + 0.624·45-s + 1.29·47-s − 0.994·49-s + 0.292·51-s + 0.968·53-s + 0.370·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8752 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8752 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 547 | \( 1 - T \) |
| good | 3 | \( 1 + 0.387T + 3T^{2} \) |
| 5 | \( 1 + 1.46T + 5T^{2} \) |
| 7 | \( 1 + 0.194T + 7T^{2} \) |
| 11 | \( 1 + 1.87T + 11T^{2} \) |
| 13 | \( 1 - 6.15T + 13T^{2} \) |
| 17 | \( 1 + 5.37T + 17T^{2} \) |
| 19 | \( 1 - 3.78T + 19T^{2} \) |
| 23 | \( 1 - 3.43T + 23T^{2} \) |
| 29 | \( 1 - 0.0833T + 29T^{2} \) |
| 31 | \( 1 - 3.95T + 31T^{2} \) |
| 37 | \( 1 - 3.24T + 37T^{2} \) |
| 41 | \( 1 + 12.5T + 41T^{2} \) |
| 43 | \( 1 + 2.32T + 43T^{2} \) |
| 47 | \( 1 - 8.88T + 47T^{2} \) |
| 53 | \( 1 - 7.04T + 53T^{2} \) |
| 59 | \( 1 + 7.89T + 59T^{2} \) |
| 61 | \( 1 + 2.95T + 61T^{2} \) |
| 67 | \( 1 - 1.41T + 67T^{2} \) |
| 71 | \( 1 - 11.1T + 71T^{2} \) |
| 73 | \( 1 + 0.418T + 73T^{2} \) |
| 79 | \( 1 + 0.962T + 79T^{2} \) |
| 83 | \( 1 - 15.7T + 83T^{2} \) |
| 89 | \( 1 + 9.44T + 89T^{2} \) |
| 97 | \( 1 - 11.6T + 97T^{2} \) |
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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
−7.48808193680598881374671081676, −6.59315126906517970063951092500, −6.14320589118724179551630922242, −5.32761102113877093454099401023, −4.68405253990287324314328123751, −3.72901480774096114006749464296, −3.21321911321867111216501909678, −2.28551173401019816040249095830, −1.06554095958181962832122961578, 0,
1.06554095958181962832122961578, 2.28551173401019816040249095830, 3.21321911321867111216501909678, 3.72901480774096114006749464296, 4.68405253990287324314328123751, 5.32761102113877093454099401023, 6.14320589118724179551630922242, 6.59315126906517970063951092500, 7.48808193680598881374671081676
