| L(s) = 1 | − 3-s − 5-s + 3·7-s + 9-s + 4·11-s + 15-s − 3·17-s + 8·19-s − 3·21-s − 23-s + 25-s − 27-s + 9·29-s + 5·31-s − 4·33-s − 3·35-s − 9·37-s + 7·41-s − 4·43-s − 45-s + 2·47-s + 2·49-s + 3·51-s + 13·53-s − 4·55-s − 8·57-s + 3·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1.13·7-s + 1/3·9-s + 1.20·11-s + 0.258·15-s − 0.727·17-s + 1.83·19-s − 0.654·21-s − 0.208·23-s + 1/5·25-s − 0.192·27-s + 1.67·29-s + 0.898·31-s − 0.696·33-s − 0.507·35-s − 1.47·37-s + 1.09·41-s − 0.609·43-s − 0.149·45-s + 0.291·47-s + 2/7·49-s + 0.420·51-s + 1.78·53-s − 0.539·55-s − 1.05·57-s + 0.390·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.028325750\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.028325750\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 9 T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 13 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 - 13 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−8.154748981038880777868297478790, −7.35431694975267318577548707562, −6.80466454616981725602709625544, −6.01223528851390510636283506656, −5.12893599196902185809428477339, −4.57855600420135458094374913034, −3.88136271853996912281498524537, −2.85499157171565939866611329878, −1.59030470424870116207137233576, −0.867145429131301457442003022812,
0.867145429131301457442003022812, 1.59030470424870116207137233576, 2.85499157171565939866611329878, 3.88136271853996912281498524537, 4.57855600420135458094374913034, 5.12893599196902185809428477339, 6.01223528851390510636283506656, 6.80466454616981725602709625544, 7.35431694975267318577548707562, 8.154748981038880777868297478790
