L(s) = 1 | − 5-s + 7-s + 4·11-s − 2·13-s − 2·17-s + 4·19-s − 23-s + 25-s + 2·29-s − 8·31-s − 35-s + 6·37-s − 2·41-s − 4·43-s + 8·47-s + 49-s + 2·53-s − 4·55-s − 4·59-s + 14·61-s + 2·65-s + 4·67-s + 2·73-s + 4·77-s + 12·83-s + 2·85-s + 6·89-s + ⋯ |
L(s) = 1 | − 0.447·5-s + 0.377·7-s + 1.20·11-s − 0.554·13-s − 0.485·17-s + 0.917·19-s − 0.208·23-s + 1/5·25-s + 0.371·29-s − 1.43·31-s − 0.169·35-s + 0.986·37-s − 0.312·41-s − 0.609·43-s + 1.16·47-s + 1/7·49-s + 0.274·53-s − 0.539·55-s − 0.520·59-s + 1.79·61-s + 0.248·65-s + 0.488·67-s + 0.234·73-s + 0.455·77-s + 1.31·83-s + 0.216·85-s + 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.585240110\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.585240110\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 + T \) |
good | 11 | \( 1 - 4 T + p T^{2} \) |
| 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 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−13.63797518051843, −13.07929001832275, −12.61479932196556, −11.98657598443403, −11.67367537960404, −11.37501274119261, −10.69959209769898, −10.25352230812036, −9.529260916755263, −9.238300805346889, −8.734997762938494, −8.148592618637886, −7.578561848145121, −7.194924008054234, −6.653686740840222, −6.116948757608631, −5.432354290817660, −4.933904510552528, −4.374397416773660, −3.749831975276111, −3.422226896473943, −2.489134169894878, −1.975179790693010, −1.165456017008614, −0.5422817550867035,
0.5422817550867035, 1.165456017008614, 1.975179790693010, 2.489134169894878, 3.422226896473943, 3.749831975276111, 4.374397416773660, 4.933904510552528, 5.432354290817660, 6.116948757608631, 6.653686740840222, 7.194924008054234, 7.578561848145121, 8.148592618637886, 8.734997762938494, 9.238300805346889, 9.529260916755263, 10.25352230812036, 10.69959209769898, 11.37501274119261, 11.67367537960404, 11.98657598443403, 12.61479932196556, 13.07929001832275, 13.63797518051843