| L(s) = 1 | + 2·3-s − 5-s − 2·7-s + 9-s − 13-s − 2·15-s + 3·17-s + 2·19-s − 4·21-s + 6·23-s − 4·25-s − 4·27-s − 29-s − 10·31-s + 2·35-s + 3·37-s − 2·39-s + 11·41-s + 12·43-s − 45-s − 10·47-s − 3·49-s + 6·51-s − 9·53-s + 4·57-s − 4·59-s − 6·61-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.447·5-s − 0.755·7-s + 1/3·9-s − 0.277·13-s − 0.516·15-s + 0.727·17-s + 0.458·19-s − 0.872·21-s + 1.25·23-s − 4/5·25-s − 0.769·27-s − 0.185·29-s − 1.79·31-s + 0.338·35-s + 0.493·37-s − 0.320·39-s + 1.71·41-s + 1.82·43-s − 0.149·45-s − 1.45·47-s − 3/7·49-s + 0.840·51-s − 1.23·53-s + 0.529·57-s − 0.520·59-s − 0.768·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 - 11 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 + T + p T^{2} \) |
| 97 | \( 1 - 11 T + p T^{2} \) |
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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
−7.61869645446882851374244373040, −7.13722684857882761955418230717, −6.09883443748694790278655067701, −5.49723602499701992739932909308, −4.47529492296087060464394190980, −3.66145277623877784495614958625, −3.14977932924638931123583315141, −2.50787058620871950908582653928, −1.38783498676763346708638910718, 0,
1.38783498676763346708638910718, 2.50787058620871950908582653928, 3.14977932924638931123583315141, 3.66145277623877784495614958625, 4.47529492296087060464394190980, 5.49723602499701992739932909308, 6.09883443748694790278655067701, 7.13722684857882761955418230717, 7.61869645446882851374244373040
