| L(s) = 1 | − 5-s + 11-s + 4·17-s + 4·19-s + 6·23-s + 25-s − 2·29-s − 6·37-s + 10·41-s − 4·43-s + 10·47-s − 7·49-s − 2·53-s − 55-s − 4·59-s − 14·61-s − 2·67-s + 4·71-s − 4·73-s + 8·79-s + 12·83-s − 4·85-s − 6·89-s − 4·95-s + 6·97-s − 14·101-s − 10·103-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.301·11-s + 0.970·17-s + 0.917·19-s + 1.25·23-s + 1/5·25-s − 0.371·29-s − 0.986·37-s + 1.56·41-s − 0.609·43-s + 1.45·47-s − 49-s − 0.274·53-s − 0.134·55-s − 0.520·59-s − 1.79·61-s − 0.244·67-s + 0.474·71-s − 0.468·73-s + 0.900·79-s + 1.31·83-s − 0.433·85-s − 0.635·89-s − 0.410·95-s + 0.609·97-s − 1.39·101-s − 0.985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.007134921\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.007134921\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−7.68955698277403015279419498543, −7.31836261654829302099336245029, −6.51946980941127769547631365002, −5.69010294548367527105261201191, −5.07377473916697763411712526988, −4.28784478551359187824963210604, −3.40787743277068203199148243778, −2.91028207780834224685672237255, −1.63466475096262592608975177673, −0.73819675070820773148377476115,
0.73819675070820773148377476115, 1.63466475096262592608975177673, 2.91028207780834224685672237255, 3.40787743277068203199148243778, 4.28784478551359187824963210604, 5.07377473916697763411712526988, 5.69010294548367527105261201191, 6.51946980941127769547631365002, 7.31836261654829302099336245029, 7.68955698277403015279419498543
