| L(s) = 1 | − 3·3-s − 7-s + 6·9-s + 5·11-s − 13-s − 4·17-s − 2·19-s + 3·21-s − 5·23-s − 5·25-s − 9·27-s + 4·29-s − 31-s − 15·33-s + 7·37-s + 3·39-s − 9·41-s + 12·43-s + 7·47-s + 49-s + 12·51-s − 4·53-s + 6·57-s + 6·59-s + 13·61-s − 6·63-s − 11·67-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 0.377·7-s + 2·9-s + 1.50·11-s − 0.277·13-s − 0.970·17-s − 0.458·19-s + 0.654·21-s − 1.04·23-s − 25-s − 1.73·27-s + 0.742·29-s − 0.179·31-s − 2.61·33-s + 1.15·37-s + 0.480·39-s − 1.40·41-s + 1.82·43-s + 1.02·47-s + 1/7·49-s + 1.68·51-s − 0.549·53-s + 0.794·57-s + 0.781·59-s + 1.66·61-s − 0.755·63-s − 1.34·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7654229595\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7654229595\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 5 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 13 T + p T^{2} \) |
| 67 | \( 1 + 11 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 - 17 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 5 T + p T^{2} \) |
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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
−9.687411640713266446124874284108, −8.916781749804225553868026019180, −7.68283143195211494191877839909, −6.63299834857139013821510766042, −6.36335980553572410729518682566, −5.54975838409236888806497825912, −4.44236143050046711072138337998, −3.91571952847248384529393635172, −2.05625495312932408576461106462, −0.68078446997238380831438979083,
0.68078446997238380831438979083, 2.05625495312932408576461106462, 3.91571952847248384529393635172, 4.44236143050046711072138337998, 5.54975838409236888806497825912, 6.36335980553572410729518682566, 6.63299834857139013821510766042, 7.68283143195211494191877839909, 8.916781749804225553868026019180, 9.687411640713266446124874284108
