| L(s) = 1 | + 4·7-s + 2·11-s − 13-s + 2·19-s + 2·23-s + 10·29-s − 4·31-s + 6·37-s + 6·41-s − 8·43-s − 12·47-s + 9·49-s − 14·53-s − 6·59-s − 2·61-s + 4·67-s + 14·73-s + 8·77-s − 12·83-s − 6·89-s − 4·91-s + 10·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 1.51·7-s + 0.603·11-s − 0.277·13-s + 0.458·19-s + 0.417·23-s + 1.85·29-s − 0.718·31-s + 0.986·37-s + 0.937·41-s − 1.21·43-s − 1.75·47-s + 9/7·49-s − 1.92·53-s − 0.781·59-s − 0.256·61-s + 0.488·67-s + 1.63·73-s + 0.911·77-s − 1.31·83-s − 0.635·89-s − 0.419·91-s + 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 187200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 187200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.716535367\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.716535367\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−13.03021129638906, −12.63487546609181, −12.16035709466825, −11.55340791479525, −11.28269125901226, −11.01628763703871, −10.22659257445530, −9.894643386021990, −9.239625689421479, −8.894314031273601, −8.100720638094721, −8.050018456178773, −7.489684840412606, −6.758229934771105, −6.449514841858050, −5.783047288464820, −5.111450704871643, −4.712825497952589, −4.469576295938858, −3.619264975462932, −3.071999327831607, −2.437073697528411, −1.657030107509092, −1.337550816246385, −0.5666746525199631,
0.5666746525199631, 1.337550816246385, 1.657030107509092, 2.437073697528411, 3.071999327831607, 3.619264975462932, 4.469576295938858, 4.712825497952589, 5.111450704871643, 5.783047288464820, 6.449514841858050, 6.758229934771105, 7.489684840412606, 8.050018456178773, 8.100720638094721, 8.894314031273601, 9.239625689421479, 9.894643386021990, 10.22659257445530, 11.01628763703871, 11.28269125901226, 11.55340791479525, 12.16035709466825, 12.63487546609181, 13.03021129638906
