| L(s) = 1 | − 2.70·3-s − 4.24·5-s + 1.27·7-s + 4.31·9-s − 0.403·11-s + 1.48·13-s + 11.4·15-s + 3.32·17-s + 6.12·19-s − 3.45·21-s − 4.20·23-s + 13.0·25-s − 3.56·27-s − 9.76·29-s + 3.32·31-s + 1.09·33-s − 5.42·35-s − 5.09·37-s − 4.02·39-s − 4.16·41-s + 9.20·43-s − 18.3·45-s + 1.54·47-s − 5.36·49-s − 8.99·51-s − 0.932·53-s + 1.71·55-s + ⋯ |
| L(s) = 1 | − 1.56·3-s − 1.89·5-s + 0.482·7-s + 1.43·9-s − 0.121·11-s + 0.412·13-s + 2.96·15-s + 0.806·17-s + 1.40·19-s − 0.754·21-s − 0.875·23-s + 2.60·25-s − 0.685·27-s − 1.81·29-s + 0.596·31-s + 0.190·33-s − 0.916·35-s − 0.836·37-s − 0.643·39-s − 0.650·41-s + 1.40·43-s − 2.73·45-s + 0.224·47-s − 0.766·49-s − 1.25·51-s − 0.128·53-s + 0.231·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 167 | \( 1 - T \) |
| good | 3 | \( 1 + 2.70T + 3T^{2} \) |
| 5 | \( 1 + 4.24T + 5T^{2} \) |
| 7 | \( 1 - 1.27T + 7T^{2} \) |
| 11 | \( 1 + 0.403T + 11T^{2} \) |
| 13 | \( 1 - 1.48T + 13T^{2} \) |
| 17 | \( 1 - 3.32T + 17T^{2} \) |
| 19 | \( 1 - 6.12T + 19T^{2} \) |
| 23 | \( 1 + 4.20T + 23T^{2} \) |
| 29 | \( 1 + 9.76T + 29T^{2} \) |
| 31 | \( 1 - 3.32T + 31T^{2} \) |
| 37 | \( 1 + 5.09T + 37T^{2} \) |
| 41 | \( 1 + 4.16T + 41T^{2} \) |
| 43 | \( 1 - 9.20T + 43T^{2} \) |
| 47 | \( 1 - 1.54T + 47T^{2} \) |
| 53 | \( 1 + 0.932T + 53T^{2} \) |
| 59 | \( 1 + 1.70T + 59T^{2} \) |
| 61 | \( 1 + 9.98T + 61T^{2} \) |
| 67 | \( 1 + 4.92T + 67T^{2} \) |
| 71 | \( 1 - 6.72T + 71T^{2} \) |
| 73 | \( 1 - 7.05T + 73T^{2} \) |
| 79 | \( 1 + 0.843T + 79T^{2} \) |
| 83 | \( 1 - 4.04T + 83T^{2} \) |
| 89 | \( 1 - 1.42T + 89T^{2} \) |
| 97 | \( 1 + 14.5T + 97T^{2} \) |
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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
−9.256303306948123213846980650274, −8.018406859114242077561790081515, −7.63992568886796312242936952869, −6.81673760914960672800102979246, −5.71356937491602369221395279874, −5.04647078548614705163857595042, −4.14871390202699848428349110623, −3.35273600522843636133450009959, −1.19159359769122529833817146915, 0,
1.19159359769122529833817146915, 3.35273600522843636133450009959, 4.14871390202699848428349110623, 5.04647078548614705163857595042, 5.71356937491602369221395279874, 6.81673760914960672800102979246, 7.63992568886796312242936952869, 8.018406859114242077561790081515, 9.256303306948123213846980650274
