| L(s) = 1 | − 2·3-s + 3·9-s + 2·11-s − 25-s − 4·27-s − 4·33-s + 2·43-s − 49-s + 2·73-s + 2·75-s + 5·81-s + 2·89-s − 97-s + 6·99-s − 2·113-s + ⋯ |
| L(s) = 1 | − 2·3-s + 3·9-s + 2·11-s − 25-s − 4·27-s − 4·33-s + 2·43-s − 49-s + 2·73-s + 2·75-s + 5·81-s + 2·89-s − 97-s + 6·99-s − 2·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7326428344\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7326428344\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 97 | \( 1 + T \) |
| good | 3 | \( ( 1 + T )^{2} \) |
| 5 | \( 1 + T^{2} \) |
| 7 | \( 1 + T^{2} \) |
| 11 | \( ( 1 - T )^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( ( 1 - T )( 1 + T ) \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( ( 1 - T )^{2} \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( ( 1 - T )^{2} \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( ( 1 - T )^{2} \) |
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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
−9.255874018551348337528311775189, −7.893489238762200979539607820241, −7.06115328162750344908809603583, −6.43401795021535655672051417549, −5.99141578892825643683592848820, −5.15927237590950332516995469464, −4.26770196557990579924853749054, −3.77939370721966531821688201664, −1.83114780771423429976051365027, −0.903292677299414520212335415425,
0.903292677299414520212335415425, 1.83114780771423429976051365027, 3.77939370721966531821688201664, 4.26770196557990579924853749054, 5.15927237590950332516995469464, 5.99141578892825643683592848820, 6.43401795021535655672051417549, 7.06115328162750344908809603583, 7.893489238762200979539607820241, 9.255874018551348337528311775189
