| L(s) = 1 | − 3·5-s + 4·7-s − 11-s + 2·13-s + 8·17-s + 6·19-s + 5·23-s + 4·25-s + 4·29-s − 31-s − 12·35-s − 3·37-s + 6·41-s − 6·43-s − 12·47-s + 9·49-s − 6·53-s + 3·55-s − 3·59-s − 6·65-s + 11·67-s − 5·71-s − 10·73-s − 4·77-s + 2·79-s + 2·83-s − 24·85-s + ⋯ |
| L(s) = 1 | − 1.34·5-s + 1.51·7-s − 0.301·11-s + 0.554·13-s + 1.94·17-s + 1.37·19-s + 1.04·23-s + 4/5·25-s + 0.742·29-s − 0.179·31-s − 2.02·35-s − 0.493·37-s + 0.937·41-s − 0.914·43-s − 1.75·47-s + 9/7·49-s − 0.824·53-s + 0.404·55-s − 0.390·59-s − 0.744·65-s + 1.34·67-s − 0.593·71-s − 1.17·73-s − 0.455·77-s + 0.225·79-s + 0.219·83-s − 2.60·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.196571249\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.196571249\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 5 | \( 1 + 3 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 8 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 11 T + p T^{2} \) |
| 71 | \( 1 + 5 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 - 5 T + p T^{2} \) |
| 97 | \( 1 - 13 T + p T^{2} \) |
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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
−7.956934014776591607573900251598, −7.61429997405739391186912692284, −6.87418988730690416783672401623, −5.68424301869154296654936031860, −5.06592095202887449192765721037, −4.53711525064349731406861939231, −3.48294779280892225932738048713, −3.09318696203506207589934529729, −1.56276573574415865649083879116, −0.856353036609411667859710468731,
0.856353036609411667859710468731, 1.56276573574415865649083879116, 3.09318696203506207589934529729, 3.48294779280892225932738048713, 4.53711525064349731406861939231, 5.06592095202887449192765721037, 5.68424301869154296654936031860, 6.87418988730690416783672401623, 7.61429997405739391186912692284, 7.956934014776591607573900251598
