| L(s) = 1 | − 3-s − 4·7-s + 9-s − 4·11-s − 6·13-s − 2·17-s − 4·19-s + 4·21-s − 27-s + 10·29-s + 4·31-s + 4·33-s + 10·37-s + 6·39-s + 2·41-s − 4·43-s + 8·47-s + 9·49-s + 2·51-s − 2·53-s + 4·57-s − 12·59-s − 10·61-s − 4·63-s + 12·67-s − 10·73-s + 16·77-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.51·7-s + 1/3·9-s − 1.20·11-s − 1.66·13-s − 0.485·17-s − 0.917·19-s + 0.872·21-s − 0.192·27-s + 1.85·29-s + 0.718·31-s + 0.696·33-s + 1.64·37-s + 0.960·39-s + 0.312·41-s − 0.609·43-s + 1.16·47-s + 9/7·49-s + 0.280·51-s − 0.274·53-s + 0.529·57-s − 1.56·59-s − 1.28·61-s − 0.503·63-s + 1.46·67-s − 1.17·73-s + 1.82·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5512247788\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5512247788\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−9.135410531389500687139112010150, −8.083031008106771138224442319061, −7.32952496038392378604851597533, −6.52019360045175260648534697692, −6.02529663612556404285575911519, −4.92657726914999322132491720707, −4.37020336744061887489402556672, −2.94506952677374126445997218043, −2.44014739617690781197152302144, −0.45364128309891857945870815661,
0.45364128309891857945870815661, 2.44014739617690781197152302144, 2.94506952677374126445997218043, 4.37020336744061887489402556672, 4.92657726914999322132491720707, 6.02529663612556404285575911519, 6.52019360045175260648534697692, 7.32952496038392378604851597533, 8.083031008106771138224442319061, 9.135410531389500687139112010150
