| L(s) = 1 | + 3-s + 3·7-s + 9-s + 5·11-s + 13-s + 5·17-s + 4·19-s + 3·21-s + 2·23-s + 27-s + 9·29-s − 3·31-s + 5·33-s − 10·37-s + 39-s − 12·41-s − 2·43-s + 9·47-s + 2·49-s + 5·51-s + 9·53-s + 4·57-s + 3·59-s + 7·61-s + 3·63-s − 9·67-s + 2·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.13·7-s + 1/3·9-s + 1.50·11-s + 0.277·13-s + 1.21·17-s + 0.917·19-s + 0.654·21-s + 0.417·23-s + 0.192·27-s + 1.67·29-s − 0.538·31-s + 0.870·33-s − 1.64·37-s + 0.160·39-s − 1.87·41-s − 0.304·43-s + 1.31·47-s + 2/7·49-s + 0.700·51-s + 1.23·53-s + 0.529·57-s + 0.390·59-s + 0.896·61-s + 0.377·63-s − 1.09·67-s + 0.240·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 62400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.671131077\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.671131077\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 + 9 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−14.28681087887925, −13.76155132050474, −13.67178953369311, −12.63027154773443, −12.13333989331284, −11.76022203371026, −11.46360699686019, −10.53494521611160, −10.26793068460898, −9.637735370286802, −8.927348050349584, −8.696509738993578, −8.130526489661827, −7.582455926295400, −6.934666442078432, −6.631713513569747, −5.703403774228124, −5.159645434592250, −4.728368318330565, −3.781505683132492, −3.607278857890822, −2.821997511833793, −1.919889426760489, −1.336872261951116, −0.8901485613640989,
0.8901485613640989, 1.336872261951116, 1.919889426760489, 2.821997511833793, 3.607278857890822, 3.781505683132492, 4.728368318330565, 5.159645434592250, 5.703403774228124, 6.631713513569747, 6.934666442078432, 7.582455926295400, 8.130526489661827, 8.696509738993578, 8.927348050349584, 9.637735370286802, 10.26793068460898, 10.53494521611160, 11.46360699686019, 11.76022203371026, 12.13333989331284, 12.63027154773443, 13.67178953369311, 13.76155132050474, 14.28681087887925
