L(s) = 1 | + 3-s + 5-s − 7-s + 9-s − 2·11-s + 15-s − 3·17-s − 19-s − 21-s + 4·23-s − 4·25-s + 27-s − 29-s − 4·31-s − 2·33-s − 35-s − 3·37-s − 7·41-s + 9·43-s + 45-s + 47-s − 6·49-s − 3·51-s + 2·53-s − 2·55-s − 57-s − 3·59-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.447·5-s − 0.377·7-s + 1/3·9-s − 0.603·11-s + 0.258·15-s − 0.727·17-s − 0.229·19-s − 0.218·21-s + 0.834·23-s − 4/5·25-s + 0.192·27-s − 0.185·29-s − 0.718·31-s − 0.348·33-s − 0.169·35-s − 0.493·37-s − 1.09·41-s + 1.37·43-s + 0.149·45-s + 0.145·47-s − 6/7·49-s − 0.420·51-s + 0.274·53-s − 0.269·55-s − 0.132·57-s − 0.390·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5568 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 29 | \( 1 + T \) |
good | 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + 7 T + p T^{2} \) |
| 43 | \( 1 - 9 T + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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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
−7.72848349557465175499031616471, −7.17978945507135438771144773415, −6.38991396091088670282194998294, −5.66001517853972341170053534435, −4.86568488345684431399643303344, −4.01784904516228166533670514383, −3.15180062711171312966538958083, −2.40278571255931783907696061948, −1.55043403927507266818260846833, 0,
1.55043403927507266818260846833, 2.40278571255931783907696061948, 3.15180062711171312966538958083, 4.01784904516228166533670514383, 4.86568488345684431399643303344, 5.66001517853972341170053534435, 6.38991396091088670282194998294, 7.17978945507135438771144773415, 7.72848349557465175499031616471