| L(s) = 1 | + 4·11-s − 2·13-s − 6·17-s − 4·19-s − 23-s − 2·29-s − 2·37-s − 10·41-s − 4·43-s − 7·49-s − 6·53-s − 4·59-s + 10·61-s − 12·67-s + 8·71-s − 10·73-s − 8·79-s + 4·83-s − 18·89-s − 2·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 1.20·11-s − 0.554·13-s − 1.45·17-s − 0.917·19-s − 0.208·23-s − 0.371·29-s − 0.328·37-s − 1.56·41-s − 0.609·43-s − 49-s − 0.824·53-s − 0.520·59-s + 1.28·61-s − 1.46·67-s + 0.949·71-s − 1.17·73-s − 0.900·79-s + 0.439·83-s − 1.90·89-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 331200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 331200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3825486075\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3825486075\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.57996783700937, −12.07197997558131, −11.74033377136356, −11.14613075400211, −10.99252217404713, −10.22679806674383, −9.882757596082566, −9.401506125385263, −8.885017099811664, −8.502654201546117, −8.185006718417264, −7.365006147987980, −6.918510627862814, −6.650061938445591, −6.125985605003468, −5.638668630291016, −4.858658385922290, −4.521359011050093, −4.127347567322694, −3.418639206820075, −3.029365985516445, −2.066356674604116, −1.919871149529107, −1.199006830100236, −0.1585725193452861,
0.1585725193452861, 1.199006830100236, 1.919871149529107, 2.066356674604116, 3.029365985516445, 3.418639206820075, 4.127347567322694, 4.521359011050093, 4.858658385922290, 5.638668630291016, 6.125985605003468, 6.650061938445591, 6.918510627862814, 7.365006147987980, 8.185006718417264, 8.502654201546117, 8.885017099811664, 9.401506125385263, 9.882757596082566, 10.22679806674383, 10.99252217404713, 11.14613075400211, 11.74033377136356, 12.07197997558131, 12.57996783700937
