| L(s) = 1 | + 3-s − 2·9-s + 4.77·11-s + 13-s − 4.77·17-s − 2.77·19-s + 1.77·23-s − 5·27-s − 9.77·29-s − 9.54·31-s + 4.77·33-s + 7.54·37-s + 39-s + 8.77·41-s − 9.77·43-s − 4·47-s − 7·49-s − 4.77·51-s + 0.227·53-s − 2.77·57-s + 7.54·59-s − 13.3·61-s − 12.7·67-s + 1.77·69-s + 3.54·71-s + 10.7·73-s + 1.77·79-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.666·9-s + 1.43·11-s + 0.277·13-s − 1.15·17-s − 0.635·19-s + 0.369·23-s − 0.962·27-s − 1.81·29-s − 1.71·31-s + 0.830·33-s + 1.24·37-s + 0.160·39-s + 1.36·41-s − 1.49·43-s − 0.583·47-s − 49-s − 0.668·51-s + 0.0313·53-s − 0.367·57-s + 0.982·59-s − 1.70·61-s − 1.56·67-s + 0.213·69-s + 0.420·71-s + 1.26·73-s + 0.199·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 - T + 3T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 - 4.77T + 11T^{2} \) |
| 17 | \( 1 + 4.77T + 17T^{2} \) |
| 19 | \( 1 + 2.77T + 19T^{2} \) |
| 23 | \( 1 - 1.77T + 23T^{2} \) |
| 29 | \( 1 + 9.77T + 29T^{2} \) |
| 31 | \( 1 + 9.54T + 31T^{2} \) |
| 37 | \( 1 - 7.54T + 37T^{2} \) |
| 41 | \( 1 - 8.77T + 41T^{2} \) |
| 43 | \( 1 + 9.77T + 43T^{2} \) |
| 47 | \( 1 + 4T + 47T^{2} \) |
| 53 | \( 1 - 0.227T + 53T^{2} \) |
| 59 | \( 1 - 7.54T + 59T^{2} \) |
| 61 | \( 1 + 13.3T + 61T^{2} \) |
| 67 | \( 1 + 12.7T + 67T^{2} \) |
| 71 | \( 1 - 3.54T + 71T^{2} \) |
| 73 | \( 1 - 10.7T + 73T^{2} \) |
| 79 | \( 1 - 1.77T + 79T^{2} \) |
| 83 | \( 1 + 0.772T + 83T^{2} \) |
| 89 | \( 1 + 5.22T + 89T^{2} \) |
| 97 | \( 1 - 14T + 97T^{2} \) |
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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
−7.910539218896715420055479959928, −7.17553451039629766401040867787, −6.36553645857486343988114827097, −5.86433141549918416476271064305, −4.80608777757039834381370764819, −3.93000076679833002891788128862, −3.42316167971233042130309041739, −2.31485011398799386639224773031, −1.56939260707687327757395777472, 0,
1.56939260707687327757395777472, 2.31485011398799386639224773031, 3.42316167971233042130309041739, 3.93000076679833002891788128862, 4.80608777757039834381370764819, 5.86433141549918416476271064305, 6.36553645857486343988114827097, 7.17553451039629766401040867787, 7.910539218896715420055479959928
