| L(s) = 1 | + 12·3-s + 88·7-s − 99·9-s + 540·11-s + 418·13-s − 594·17-s + 836·19-s + 1.05e3·21-s + 4.10e3·23-s − 4.10e3·27-s − 594·29-s + 4.25e3·31-s + 6.48e3·33-s + 298·37-s + 5.01e3·39-s + 1.72e4·41-s + 1.21e4·43-s + 1.29e3·47-s − 9.06e3·49-s − 7.12e3·51-s − 1.94e4·53-s + 1.00e4·57-s − 7.66e3·59-s − 3.47e4·61-s − 8.71e3·63-s − 2.18e4·67-s + 4.92e4·69-s + ⋯ |
| L(s) = 1 | + 0.769·3-s + 0.678·7-s − 0.407·9-s + 1.34·11-s + 0.685·13-s − 0.498·17-s + 0.531·19-s + 0.522·21-s + 1.61·23-s − 1.08·27-s − 0.131·29-s + 0.795·31-s + 1.03·33-s + 0.0357·37-s + 0.528·39-s + 1.60·41-s + 0.997·43-s + 0.0855·47-s − 0.539·49-s − 0.383·51-s − 0.953·53-s + 0.408·57-s − 0.286·59-s − 1.19·61-s − 0.276·63-s − 0.593·67-s + 1.24·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.680835047\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.680835047\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 4 p T + p^{5} T^{2} \) |
| 7 | \( 1 - 88 T + p^{5} T^{2} \) |
| 11 | \( 1 - 540 T + p^{5} T^{2} \) |
| 13 | \( 1 - 418 T + p^{5} T^{2} \) |
| 17 | \( 1 + 594 T + p^{5} T^{2} \) |
| 19 | \( 1 - 44 p T + p^{5} T^{2} \) |
| 23 | \( 1 - 4104 T + p^{5} T^{2} \) |
| 29 | \( 1 + 594 T + p^{5} T^{2} \) |
| 31 | \( 1 - 4256 T + p^{5} T^{2} \) |
| 37 | \( 1 - 298 T + p^{5} T^{2} \) |
| 41 | \( 1 - 17226 T + p^{5} T^{2} \) |
| 43 | \( 1 - 12100 T + p^{5} T^{2} \) |
| 47 | \( 1 - 1296 T + p^{5} T^{2} \) |
| 53 | \( 1 + 19494 T + p^{5} T^{2} \) |
| 59 | \( 1 + 7668 T + p^{5} T^{2} \) |
| 61 | \( 1 + 34738 T + p^{5} T^{2} \) |
| 67 | \( 1 + 21812 T + p^{5} T^{2} \) |
| 71 | \( 1 + 46872 T + p^{5} T^{2} \) |
| 73 | \( 1 + 67562 T + p^{5} T^{2} \) |
| 79 | \( 1 + 76912 T + p^{5} T^{2} \) |
| 83 | \( 1 + 67716 T + p^{5} T^{2} \) |
| 89 | \( 1 - 29754 T + p^{5} T^{2} \) |
| 97 | \( 1 - 122398 T + p^{5} T^{2} \) |
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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
−13.12400555138146720027059614311, −11.70795994597655278479936789499, −10.97627873869666442011833267347, −9.284121033467762187606371672735, −8.698285826539357191504504361104, −7.46466515461146922442899717475, −6.05846110235853288020527064849, −4.40870254790742148245276752075, −3.00679451853347636083245749614, −1.31952607353875510623433652554,
1.31952607353875510623433652554, 3.00679451853347636083245749614, 4.40870254790742148245276752075, 6.05846110235853288020527064849, 7.46466515461146922442899717475, 8.698285826539357191504504361104, 9.284121033467762187606371672735, 10.97627873869666442011833267347, 11.70795994597655278479936789499, 13.12400555138146720027059614311
