| L(s) = 1 | + 3-s − 2.38·7-s + 9-s − 3.47·11-s − 5·13-s + 4.85·17-s + 6.70·19-s − 2.38·21-s − 5.47·23-s + 27-s − 8.23·29-s − 6.85·31-s − 3.47·33-s − 3·37-s − 5·39-s + 5.32·41-s − 11.8·43-s − 9·47-s − 1.32·49-s + 4.85·51-s − 5.09·53-s + 6.70·57-s + 0.381·59-s + 1.14·61-s − 2.38·63-s + 4.94·67-s − 5.47·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.900·7-s + 0.333·9-s − 1.04·11-s − 1.38·13-s + 1.17·17-s + 1.53·19-s − 0.519·21-s − 1.14·23-s + 0.192·27-s − 1.52·29-s − 1.23·31-s − 0.604·33-s − 0.493·37-s − 0.800·39-s + 0.831·41-s − 1.80·43-s − 1.31·47-s − 0.189·49-s + 0.679·51-s − 0.699·53-s + 0.888·57-s + 0.0497·59-s + 0.146·61-s − 0.300·63-s + 0.604·67-s − 0.658·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1500 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 2.38T + 7T^{2} \) |
| 11 | \( 1 + 3.47T + 11T^{2} \) |
| 13 | \( 1 + 5T + 13T^{2} \) |
| 17 | \( 1 - 4.85T + 17T^{2} \) |
| 19 | \( 1 - 6.70T + 19T^{2} \) |
| 23 | \( 1 + 5.47T + 23T^{2} \) |
| 29 | \( 1 + 8.23T + 29T^{2} \) |
| 31 | \( 1 + 6.85T + 31T^{2} \) |
| 37 | \( 1 + 3T + 37T^{2} \) |
| 41 | \( 1 - 5.32T + 41T^{2} \) |
| 43 | \( 1 + 11.8T + 43T^{2} \) |
| 47 | \( 1 + 9T + 47T^{2} \) |
| 53 | \( 1 + 5.09T + 53T^{2} \) |
| 59 | \( 1 - 0.381T + 59T^{2} \) |
| 61 | \( 1 - 1.14T + 61T^{2} \) |
| 67 | \( 1 - 4.94T + 67T^{2} \) |
| 71 | \( 1 - 0.326T + 71T^{2} \) |
| 73 | \( 1 - 7.85T + 73T^{2} \) |
| 79 | \( 1 - 9.70T + 79T^{2} \) |
| 83 | \( 1 + 12.3T + 83T^{2} \) |
| 89 | \( 1 - 9.47T + 89T^{2} \) |
| 97 | \( 1 + 0.0901T + 97T^{2} \) |
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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
−9.457636190321296852113351308065, −8.019178104346953394593628703447, −7.64607371416913566428889752093, −6.85469814798855402485519664072, −5.60423788819000350154116686817, −5.06529075712942192961563812018, −3.61894689732592231853074951739, −3.04214794280081466664495102363, −1.91602250901553934944986212385, 0,
1.91602250901553934944986212385, 3.04214794280081466664495102363, 3.61894689732592231853074951739, 5.06529075712942192961563812018, 5.60423788819000350154116686817, 6.85469814798855402485519664072, 7.64607371416913566428889752093, 8.019178104346953394593628703447, 9.457636190321296852113351308065
