| L(s) = 1 | + 4·11-s − 13-s + 6·17-s − 3·19-s − 6·23-s − 3·37-s + 4·41-s − 6·47-s + 6·53-s + 2·59-s + 11·61-s − 7·67-s − 4·71-s + 11·73-s − 15·79-s − 6·83-s − 14·89-s − 97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 1.20·11-s − 0.277·13-s + 1.45·17-s − 0.688·19-s − 1.25·23-s − 0.493·37-s + 0.624·41-s − 0.875·47-s + 0.824·53-s + 0.260·59-s + 1.40·61-s − 0.855·67-s − 0.474·71-s + 1.28·73-s − 1.68·79-s − 0.658·83-s − 1.48·89-s − 0.101·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 176400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 176400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.403565526\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.403565526\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 - 11 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 + 15 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−13.13118098030414, −12.58735229097254, −12.26206377332416, −11.74929426598989, −11.47660292321023, −10.80178781569904, −10.23871052140934, −9.838375120538697, −9.516585923835574, −8.859621718606971, −8.291663859655099, −8.077992244864879, −7.233632889556501, −6.986458234381251, −6.354267782031527, −5.719506790918576, −5.568385497332341, −4.636444492799353, −4.206887291652207, −3.696828253640672, −3.184848859206841, −2.442361209771827, −1.780268039791769, −1.246400273325661, −0.4620324961338369,
0.4620324961338369, 1.246400273325661, 1.780268039791769, 2.442361209771827, 3.184848859206841, 3.696828253640672, 4.206887291652207, 4.636444492799353, 5.568385497332341, 5.719506790918576, 6.354267782031527, 6.986458234381251, 7.233632889556501, 8.077992244864879, 8.291663859655099, 8.859621718606971, 9.516585923835574, 9.838375120538697, 10.23871052140934, 10.80178781569904, 11.47660292321023, 11.74929426598989, 12.26206377332416, 12.58735229097254, 13.13118098030414
