| L(s) = 1 | + 3-s − 2·9-s + 2·11-s + 4·17-s + 2·19-s − 23-s − 5·27-s + 9·29-s − 4·31-s + 2·33-s − 4·37-s − 41-s − 9·43-s + 4·51-s + 10·53-s + 2·57-s + 10·59-s − 9·61-s − 5·67-s − 69-s + 14·71-s + 12·73-s + 14·79-s + 81-s + 11·83-s + 9·87-s + 15·89-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 2/3·9-s + 0.603·11-s + 0.970·17-s + 0.458·19-s − 0.208·23-s − 0.962·27-s + 1.67·29-s − 0.718·31-s + 0.348·33-s − 0.657·37-s − 0.156·41-s − 1.37·43-s + 0.560·51-s + 1.37·53-s + 0.264·57-s + 1.30·59-s − 1.15·61-s − 0.610·67-s − 0.120·69-s + 1.66·71-s + 1.40·73-s + 1.57·79-s + 1/9·81-s + 1.20·83-s + 0.964·87-s + 1.58·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.551664273\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.551664273\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + T + p T^{2} \) |
| 43 | \( 1 + 9 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 9 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 - 14 T + p T^{2} \) |
| 73 | \( 1 - 12 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 - 11 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−8.020571494984061845867346475747, −6.90719983642993852285618005656, −6.46486049788116112139876619710, −5.48590456866964101803135474028, −5.08435240280319951864397004172, −3.94107483858237389403954328536, −3.42258152936512899074803603570, −2.69335986803017718150953203417, −1.78836787573733537068218027473, −0.74376440164816432305546696791,
0.74376440164816432305546696791, 1.78836787573733537068218027473, 2.69335986803017718150953203417, 3.42258152936512899074803603570, 3.94107483858237389403954328536, 5.08435240280319951864397004172, 5.48590456866964101803135474028, 6.46486049788116112139876619710, 6.90719983642993852285618005656, 8.020571494984061845867346475747
