| L(s) = 1 | + 2·3-s + 7-s + 9-s − 4·11-s + 4·13-s + 2·17-s − 6·19-s + 2·21-s − 5·25-s − 4·27-s − 2·29-s − 4·31-s − 8·33-s + 10·37-s + 8·39-s − 10·41-s + 4·43-s + 4·47-s + 49-s + 4·51-s − 2·53-s − 12·57-s − 10·59-s − 8·61-s + 63-s − 8·67-s − 6·73-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.377·7-s + 1/3·9-s − 1.20·11-s + 1.10·13-s + 0.485·17-s − 1.37·19-s + 0.436·21-s − 25-s − 0.769·27-s − 0.371·29-s − 0.718·31-s − 1.39·33-s + 1.64·37-s + 1.28·39-s − 1.56·41-s + 0.609·43-s + 0.583·47-s + 1/7·49-s + 0.560·51-s − 0.274·53-s − 1.58·57-s − 1.30·59-s − 1.02·61-s + 0.125·63-s − 0.977·67-s − 0.702·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 236992 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 236992 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.025874595\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.025874595\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−13.10538504700834, −12.56588706118657, −12.05063050051957, −11.40339317328516, −10.97847143709948, −10.58141458073575, −10.17554533201752, −9.404527170870173, −9.199273422656885, −8.585223177132431, −8.142198430445182, −7.883942041417186, −7.502237678727183, −6.806019287673297, −5.977174630065482, −5.898817955511504, −5.196792518742048, −4.473242290197708, −4.065730834436870, −3.493135133143293, −2.974674524561930, −2.454165471222341, −1.876375417950735, −1.421621868139960, −0.3354250938591134,
0.3354250938591134, 1.421621868139960, 1.876375417950735, 2.454165471222341, 2.974674524561930, 3.493135133143293, 4.065730834436870, 4.473242290197708, 5.196792518742048, 5.898817955511504, 5.977174630065482, 6.806019287673297, 7.502237678727183, 7.883942041417186, 8.142198430445182, 8.585223177132431, 9.199273422656885, 9.404527170870173, 10.17554533201752, 10.58141458073575, 10.97847143709948, 11.40339317328516, 12.05063050051957, 12.56588706118657, 13.10538504700834
