L(s) = 1 | − 7-s − 5·11-s − 13-s + 3·17-s + 6·19-s + 6·23-s + 9·29-s − 6·37-s − 8·41-s + 6·43-s − 3·47-s + 49-s − 12·53-s + 8·59-s − 4·61-s − 4·67-s + 8·71-s − 10·73-s + 5·77-s + 3·79-s + 12·83-s + 16·89-s + 91-s − 7·97-s + 101-s + 103-s + 107-s + ⋯ |
L(s) = 1 | − 0.377·7-s − 1.50·11-s − 0.277·13-s + 0.727·17-s + 1.37·19-s + 1.25·23-s + 1.67·29-s − 0.986·37-s − 1.24·41-s + 0.914·43-s − 0.437·47-s + 1/7·49-s − 1.64·53-s + 1.04·59-s − 0.512·61-s − 0.488·67-s + 0.949·71-s − 1.17·73-s + 0.569·77-s + 0.337·79-s + 1.31·83-s + 1.69·89-s + 0.104·91-s − 0.710·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.770680619\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.770680619\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 3 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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
−15.55055519395035, −14.84908252267391, −14.21412134632405, −13.74604457509140, −13.19422297591257, −12.74339989713207, −12.06136403779937, −11.77070273918895, −10.82080241817235, −10.48980720352163, −9.940539189115425, −9.403778445929913, −8.753990480050841, −8.011207295559557, −7.665308106261814, −6.981248644955956, −6.432839982493328, −5.540685292663391, −5.104504721107208, −4.695092155201445, −3.489478575719230, −3.072206370497234, −2.509396087950467, −1.410138384852940, −0.5469531897153716,
0.5469531897153716, 1.410138384852940, 2.509396087950467, 3.072206370497234, 3.489478575719230, 4.695092155201445, 5.104504721107208, 5.540685292663391, 6.432839982493328, 6.981248644955956, 7.665308106261814, 8.011207295559557, 8.753990480050841, 9.403778445929913, 9.940539189115425, 10.48980720352163, 10.82080241817235, 11.77070273918895, 12.06136403779937, 12.74339989713207, 13.19422297591257, 13.74604457509140, 14.21412134632405, 14.84908252267391, 15.55055519395035