L(s) = 1 | − 3-s + 2·5-s − 4·7-s + 9-s − 2·13-s − 2·15-s + 2·17-s − 4·19-s + 4·21-s − 25-s − 27-s − 2·29-s + 8·31-s − 8·35-s − 2·37-s + 2·39-s + 10·41-s − 4·43-s + 2·45-s + 9·49-s − 2·51-s − 6·53-s + 4·57-s − 12·59-s − 2·61-s − 4·63-s − 4·65-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.894·5-s − 1.51·7-s + 1/3·9-s − 0.554·13-s − 0.516·15-s + 0.485·17-s − 0.917·19-s + 0.872·21-s − 1/5·25-s − 0.192·27-s − 0.371·29-s + 1.43·31-s − 1.35·35-s − 0.328·37-s + 0.320·39-s + 1.56·41-s − 0.609·43-s + 0.298·45-s + 9/7·49-s − 0.280·51-s − 0.824·53-s + 0.529·57-s − 1.56·59-s − 0.256·61-s − 0.503·63-s − 0.496·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25392 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.020265618\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.020265618\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 23 | \( 1 \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 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 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−15.63411957642595, −14.83937665924698, −14.12658769372313, −13.74227223670801, −12.97614324157846, −12.83529920515560, −12.16685052780274, −11.73804659362467, −10.73592985181302, −10.51438254296728, −9.704627650448135, −9.609338273878400, −8.993057361511326, −8.068261765942283, −7.491112457462473, −6.626236464855135, −6.356567996067872, −5.893000264138025, −5.209870270293870, −4.493756588470493, −3.758957574838823, −2.951011667081337, −2.376204169947791, −1.467110359045164, −0.4105440636776178,
0.4105440636776178, 1.467110359045164, 2.376204169947791, 2.951011667081337, 3.758957574838823, 4.493756588470493, 5.209870270293870, 5.893000264138025, 6.356567996067872, 6.626236464855135, 7.491112457462473, 8.068261765942283, 8.993057361511326, 9.609338273878400, 9.704627650448135, 10.51438254296728, 10.73592985181302, 11.73804659362467, 12.16685052780274, 12.83529920515560, 12.97614324157846, 13.74227223670801, 14.12658769372313, 14.83937665924698, 15.63411957642595