| L(s) = 1 | + 2.79·3-s + 0.208·7-s + 4.79·9-s − 11-s + 13-s − 0.791·17-s + 6.58·19-s + 0.582·21-s + 3.79·23-s + 4.99·27-s + 6.79·29-s − 8.58·31-s − 2.79·33-s + 2.58·37-s + 2.79·39-s − 1.41·41-s + 10·43-s + 1.41·47-s − 6.95·49-s − 2.20·51-s − 11.3·53-s + 18.3·57-s − 10.5·59-s + 4.20·61-s + 0.999·63-s + 4·67-s + 10.5·69-s + ⋯ |
| L(s) = 1 | + 1.61·3-s + 0.0788·7-s + 1.59·9-s − 0.301·11-s + 0.277·13-s − 0.191·17-s + 1.51·19-s + 0.127·21-s + 0.790·23-s + 0.962·27-s + 1.26·29-s − 1.54·31-s − 0.485·33-s + 0.424·37-s + 0.446·39-s − 0.221·41-s + 1.52·43-s + 0.206·47-s − 0.993·49-s − 0.309·51-s − 1.56·53-s + 2.43·57-s − 1.37·59-s + 0.538·61-s + 0.125·63-s + 0.488·67-s + 1.27·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.931845190\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.931845190\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 - 2.79T + 3T^{2} \) |
| 7 | \( 1 - 0.208T + 7T^{2} \) |
| 13 | \( 1 - T + 13T^{2} \) |
| 17 | \( 1 + 0.791T + 17T^{2} \) |
| 19 | \( 1 - 6.58T + 19T^{2} \) |
| 23 | \( 1 - 3.79T + 23T^{2} \) |
| 29 | \( 1 - 6.79T + 29T^{2} \) |
| 31 | \( 1 + 8.58T + 31T^{2} \) |
| 37 | \( 1 - 2.58T + 37T^{2} \) |
| 41 | \( 1 + 1.41T + 41T^{2} \) |
| 43 | \( 1 - 10T + 43T^{2} \) |
| 47 | \( 1 - 1.41T + 47T^{2} \) |
| 53 | \( 1 + 11.3T + 53T^{2} \) |
| 59 | \( 1 + 10.5T + 59T^{2} \) |
| 61 | \( 1 - 4.20T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 + 10.7T + 71T^{2} \) |
| 73 | \( 1 - 7.79T + 73T^{2} \) |
| 79 | \( 1 + 15.5T + 79T^{2} \) |
| 83 | \( 1 - 9.95T + 83T^{2} \) |
| 89 | \( 1 + 0.791T + 89T^{2} \) |
| 97 | \( 1 - 6.20T + 97T^{2} \) |
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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
−9.443059319231059430533697548768, −9.179444312415244742936704702838, −8.157013255232216628929813430220, −7.63909396369849420762098034312, −6.78923680454824226029313914357, −5.49230567504320095418253672594, −4.42142255016391846282589323249, −3.34898143077619068926443810317, −2.71115258607888438874401649931, −1.42776846152052937719727263869,
1.42776846152052937719727263869, 2.71115258607888438874401649931, 3.34898143077619068926443810317, 4.42142255016391846282589323249, 5.49230567504320095418253672594, 6.78923680454824226029313914357, 7.63909396369849420762098034312, 8.157013255232216628929813430220, 9.179444312415244742936704702838, 9.443059319231059430533697548768
