| L(s) = 1 | + 3-s − 2·9-s + 11-s + 13-s + 6·17-s + 2·19-s + 6·23-s − 5·25-s − 5·27-s + 9·29-s − 4·31-s + 33-s + 2·37-s + 39-s + 6·41-s + 4·43-s − 6·47-s + 6·51-s + 2·57-s − 3·59-s − 11·61-s − 11·67-s + 6·69-s − 2·73-s − 5·75-s − 5·79-s + 81-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 2/3·9-s + 0.301·11-s + 0.277·13-s + 1.45·17-s + 0.458·19-s + 1.25·23-s − 25-s − 0.962·27-s + 1.67·29-s − 0.718·31-s + 0.174·33-s + 0.328·37-s + 0.160·39-s + 0.937·41-s + 0.609·43-s − 0.875·47-s + 0.840·51-s + 0.264·57-s − 0.390·59-s − 1.40·61-s − 1.34·67-s + 0.722·69-s − 0.234·73-s − 0.577·75-s − 0.562·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.671518327\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.671518327\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + 11 T + p T^{2} \) |
| 67 | \( 1 + 11 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 5 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 - 13 T + p T^{2} \) |
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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
−7.75072952617112506252815171486, −7.33565854633915742471869024800, −6.26961612187096828310641750835, −5.79422409173949635634750303522, −5.01452496008455191051435135227, −4.16389092300858550844971172336, −3.19383653998384407255543003178, −2.95292059023757541625850002671, −1.75230157949516182028394151559, −0.796724616371184391113970350001,
0.796724616371184391113970350001, 1.75230157949516182028394151559, 2.95292059023757541625850002671, 3.19383653998384407255543003178, 4.16389092300858550844971172336, 5.01452496008455191051435135227, 5.79422409173949635634750303522, 6.26961612187096828310641750835, 7.33565854633915742471869024800, 7.75072952617112506252815171486
